I'm not sure if this will still be helpful, but here is my understanding of the Quillen model. Everything correct that I write below, I learned from John Francis. (Probably in the same lecture that Theo mentioned in his comment above.) Any mistakes are not his fault---more likely an error in my understanding.
Before we begin: Quillen v Sullivan.
As others have mentioned, Quillen gets you a DG Lie algebra, where as the Sullivan model will get you a commutative DG algebra. As you write, the passage from one to the other is (almost) Koszul duality. Really, a Lie algebra will get you a co-commutative coalgebra by Koszul duality, and a commutative algebra will get you a coLie algebra. You can bridge the world of coalgebras and algebras when you have some finiteness conditions--for instance, if the rational homotopy groups are finite-dimensional in each degree. Then you can simply take linear duals to get from coalgebras to algebras.
A way to find Lie algebras.
So where do (DG) Lie algebras come from? There is a natural place that one finds Lie algebras, before knowing about the Quillen model: Lie algebras arise as the tangent space (at the identity) of a Lie group $G$.
Now, if you're an algebraist, you might claim another origin of Lie algebras: If you have any kind of Hopf algebra, you can look at the primitives of the Hopf algebra. These always form a Lie algebra.
(Recall that a Hopf algebra has a coproduct $\Delta: H \to H \otimes H$, and a primitive of $H$ is defined to be an element $x$ such that $\Delta(x) = 1 \otimes x + x \otimes 1.$)
One link between the algebraist's fountain of Lie algebras, and the geometer's, is that many Hopf algebras arise as functions on finite groups. If you are well-versed in algebra, one natural place to find Lie algebras, then, would be to take a finite group, take functions on that group, then take primitives.
A cooler link arises when a geometer looks at distributions near the identity of $G$ (which are dual to 'functions on $G$') rather than functions themselves. This isn't so obviously the right thing to look at in the finite groups example, but if you believe that functions on a Lie group $G$ are like de Rham forms on $G$, then you'd believe that something like 'the duals to functions on $G$' (which are closer to vector fields) would somehow safeguard the Lie algebra structure. The point being, you should expect to find Lie structures to arise from things that look like 'duals to functions on a group'. So one should take 'distributions' to be the Hopf algebra in question, and look at its primitives to find the Lie algebra of 'vector fields.'
A (fantastical) summary of the Quillen model.
Let us assume for a moment that your space $X$ happens to equal $BG$ for some Lie group, and you want to make a Lie algebra out of it. Then, by the above, what you could do is take $\Omega X = \Omega B G = G$, then look at the primitives of the Hopf algebra known as `distributions on $\Omega X$'.
Now, instead of considering just Lie groups, let's believe in a fantasy world (later made reality) in which all the heuristics I outlined for a Lie group $G$ will also work for a based loop space $\Omega Y$. A loop space is `like a group' because it has a space of multiplications, all invertible (up to homotopy). Moreover, any space $X$ is the $B$ (classifying space) of a loop space--namely, $X \cong B \Omega X$. So this will give us a way to associate a Lie algebra to any space, if you believe in the fantasy.
Blindly following the analogy, `functions on $\Omega X$' is like cochains on $\Omega X$, and the dual to this (i.e., distributions) is now chains on $\Omega X$. That is, $C_\bullet \Omega X$ should have the structure of what looks like a Hopf algebra. And its primitives should be the Lie algebra you're looking for.
What Quillen Does.
So if that's the story, what else is there? Of course, there is the fantasy, which I have to explain. Loop spaces are most definitely not Lie groups. Their products have $A_\infty$ structure, and correspondigly, we should be talking about things like homotopy Hopf algebras, not Hopf algebras on the nose. What Quillen does is not to take care of all the coherence issues, but to change the models of the objects he's working with.
For instance, one can get an actual simplicial group out of a space $X$ by Kan's construction $G$. This is a model for the loop space $\Omega X$, and this is what Quillen looks at instead of looking only at $\Omega X$, which is too flimsy. From this, taking group algebras over $\mathbb{Q}$ and completing (these are the simplicial chains, i.e., distributions), he obtains completed simplicial Hopf algebras. Again, instead of trying to make my fantasy precise in a world where one has to deal with higher algebraic structures (homotopy up to homotopy, et cetera) he uses this nice simplicial model. To complete the story, he takes level-wise primitives, obtaining DG Lie algebras.
Edit: This is from Tom's comment below. To recover a $k$-connected group or a $k$-connected Lie algebra from the associated $k$-connected complete Hopf algebra, you need $k \geq 0$. And $k$-connected groups correspond to $k+1$-connected spaces. This is why you need simply connected spaces in the equivalence.
I'm not sure I gave any 'high concept' as to 'why Quillen's construction works', but this is at least a road map I can remember.
$\newcommand\Z{\mathbf{Z}}$
$\newcommand\Q{\mathbf{Q}}$
I'm a number theorist who already thinks of the algebraic $K$-theory of $\Z$ as part of number theory anyway, but let me make some general remarks.
A narrow answer: Since (following work of Voevodsky, Rost, and many others) the $K$-groups of $\Z$ may be identified with Galois cohomology groups (with controlled ramification) of certain Tate twists $\Z_p(n)$, the answer is literally "the information contained in the $K$-groups is the same as the information contained in the appropriate Galois cohomology groups."
To make this more specific, one can look at the rank and the torsion part of these groups.
The ranks (of the odd $K$-groups) are related to $H^1(\Q,\Q_p(n))$ (the Galois groups will be modified by local conditions which I will suppress), which is related to the group of extensions of (the Galois modules) $\Q_p$ by $\Q_p(n)$. A formula of Tate computes the Euler characteristic of $\Q_p(n)$, but the cohomological dimension of $\Q$ is $2$, so there is also an $H^2$ term. The computation of the rational $K$-groups by Borel, together with the construction of surjective Chern classes by Soulé allows one to compute these groups explicitly for positive integers $n$. There is no other proof of this result, as far as I know (of course it is trivial in the case when $p$ is regular).
The (interesting) torsion classes in $K$-groups are directly related to the class groups of cyclotomic extensions. For example, let $\chi: \mathrm{Gal}(\overline{\Q}/\Q) \rightarrow \mathbf{F}^{\times}_p$ be the mod-$p$ cyclotomic character. Then one can ask whether there exist extensions of Galois modules:
$$0 \rightarrow \mathbf{F}_p(\chi^{2n}) \rightarrow V \rightarrow \mathbf{F}_p \rightarrow 0$$
which are unramified everywhere. Such classes (warning: possible sign error/indexing disaster alert) are the same as giving $p$-torsion classes in $K_{4n}(\Z)$. The non-existence of such classes for all $n$ and $p$ is Vandiver's conjecture. Now we see that:
The finiteness of $K$-groups implies that, for any fixed $n$, there are only finitely many $p$ such that an extension exists. An, for example, an explicit computation of $K_8(\Z)$ will determine explicitly all such primes (namely, the primes dividing the order of $K_8(\Z)$). As a number theorist, I think that Vandiver's conjecture is a little silly --- its natural generalization is false and there's no compelling reason for it to be true. The "true" statement which is always correct is that $K_{2n}(\mathcal{O}_F)$ is finite.
Regulators. Also important is that $K_*(\Z)$ admits natural maps to real vector spaces whose image is (in many cases) a lattice whose volume can be given in terms of zeta functions (Borel). So $K$-theory is directly related to problems concerning zeta values, which are surely of interest to number theorists. The natural generalization of this conjecture is one of the fundamental problems of number theory (and includes as special cases the Birch--Swinnerton-Dyer conjecture, etc.). There are also $p$-adic versions of these constructions which also immediately lead to open problems, even for $K_1$ (specifically, Leopoldt's conjecture and its generalizations.)
A broader answer: A lot of number theorists are interested in the Langlands programme, and in particular with automorphic representations for $\mathrm{GL}(n)/\Q$. There is a special subclass of such representations (regular, algebraic, and cuspidal) which on the one hand give rise to regular $n$-dimensional geometric Galois representations (which should be irreducible and motivic), and on the other hand correspond to rational cohomology classes in the symmetric space for $\mathrm{GL}(n)/\Q$, which (as it is essentially a $K(\pi,1)$) is the same as the rational cohomology of congruence subgroups of $\mathrm{GL}_n(\Z)$. Recent experience suggests that in order to prove reciprocity conjectures it will also be necessary to understand the integral cohomology of these groups. Now the cohomology classes corresponding to these cuspidal forms are unstable classes, but one can imagine a square with four corners as follows:
stable cohomology over $\mathbf{R}$: the trivial representation.
unstable cohomology over $\mathbf{R}$: regular algebraic automorphic forms for $\mathrm{GL}(n)/\Q$.
stable cohomology over $\mathbf{Z}$: algebraic $K$-theory.
unstable cohomology over $\mathbf{Z}$: ?"torsion automorphic forms"?, or at the very least, something interesting and important but not well understood.
From this optic, algebraic $K$-theory of (say) rings of integers of number fields is very naturally part of the Langlands programme, broadly construed.
Final Remark: algebraic K-theory is a (beautiful) language invented by Quillen to explain certain phenomena; I think it is a little dangerous to think of it as being an application of "homotopy theory". Progress in the problems above required harmonic analysis and representation theory (understanding automorphic forms), Galois cohomology, as well as homotopy theory and many other ingredients. Progress in open questions (such as Leopoldt's conjecture) will also presumably require completely new methods.
Best Answer
Good question!
Actually, it seems unlikely that perfectoid methods per se play a key role in homotopy theory. The reason is that perfectoid things are "infinitely ramified", but there are theorems to the effect that many objects of interest in algebraic topology do not admit any ramified covers. For example, for a $K(n)$-local $E_\infty$-ring $A$, the ring $\pi_0 A$ can never be a ramified extension of $\mathbb Z_p$. For $n=1$, this follows from $\pi_0 A$ carrying a canonical structure of a $\delta$-ring.
On the other hand, it seems that the more recent prismatic ideas have a chance of being more directly of interest. One point of overlap is that both $K(1)$-local $E_\infty$-rings and prismatic things are very closely linked to $\delta$-rings. Another point of overlap is that computations in prismatic cohomology are often done in some form via analysis of Drinfeld's stack $\Sigma$, and some kind of descent. This can often be mimicked in algebraic topology by using relative $\mathrm{THH}$ and some kind of Adams spectral sequence. See for example the work of Liu--Wang computing $\mathrm{TC}$ of rings of integers of $p$-adic fields (reproving the results of Hesselholt--Madsen). Another application of prismatic cohomology is the work of Bhatt--Clausen--Mathew showing that $L_{K(1)}K(\mathbb Z/p^n\mathbb Z)$ vanishes (since reproved by Land--Mathew--Meier--Tamme by other means -- and better, as their result applies to all chromatic heights). The idea of using relative $\mathrm{THH}$ has also (in a slightly different context) been used by Hahn--Wilson to study redshift.
So this is all in the spirit of 1) in your question. By the way, here's a weird conjecture about the relation between prisms and algebraic topology. [Edit: This conjecture is wrong. See Jacob Lurie's comments below.] Recall that perfect prisms $(A,I)$ are equivalent to perfectoid rings $R=A/I$, and for such the $p$-completed $\mathrm{THH}$ defines a cyclotomic spectrum concentrated in even degrees with $\pi_\ast \mathrm{THH}(R;\mathbb Z_p) = I^{\ast/2}/I^{\ast/2+1}$. I conjecture that this process can be extended to all prisms, functorially defining a cyclotomic spectrum (concentrated in even degrees) with $\pi_\ast = I^{\ast/2}/I^{\ast/2+1}$ for any prism $(A,I)$. For example, for a Breuil--Kisin prism $(\mathfrak S=W(k)[[u]],I)$ with $\mathcal O_K=\mathfrak S/I$, this ought to give the relative $\mathrm{THH}(\mathcal O_K/\mathbb S[u];\mathbb Z_p)$.
Such a construction would induce some interesting arithmetic structures on any prism. For example, by taking $S^1$-fixed points, one should get an even $E_\infty$-ring with $\pi_0 = A$, and $\pi_2$ some invertible $A$-module; the latter should be the Breuil--Kisin twist $A\{1\}$. Moreover, this even $E_\infty$-ring gives a $1$-dimensional formal group over $A$. Such a natural $1$-dimensional formal group over any prism has actually been constructed by Drinfeld (but I still haven't fully understood its significance).
Regarding 2), 3) and 4), I don't really know. I think a key open question is the computation of the Picard group of the $K(n)$-local category, which is related to the $\mathcal O_D^\times$-equivariant line bundles on the Lubin--Tate space. It's not inconceivable that perfectoid methods can be helpful here!