One thing to keep in mind is that the process that starts with LAGs and ends up with root systems is not "functorial". To keep everything very definite, let me restrict attention to finite-dimensional semisimple Lie algebras over $\mathbb C$. Then there are maps {isomorphism classes of s.s. complex Lie algebras} ↔ {isomorphism classes of Dynkin diagrams}, and at the level of isomorphism classes, the two maps are inverse to each other. In fact, there is a wonderfully functorial map going ←, i.e. from Dynkin diagrams to algebras, which was worked out by Serre, I think (maybe Chevalley). But the → map requires making all sorts of choices: pick a Cartan subalgebra, pick a notion of "positive" for it. Let $\mathfrak g$ be the Lie algebra and $\mathfrak h$ the chosen Cartan; then the group $\operatorname{Aut}(\mathfrak g)$ acts transitively on choices of simple system, and the stabilizer is precisely $\mathfrak h$. (Or, rather, in $\operatorname{Aut}(\mathfrak g)$ there are "inner" automorphism and "outer" automorphisms, and the inner ones in fact act transitively, and $\operatorname{Out}/\operatorname{Inn}$ acts as non-trivial diagram automorphisms. By "precisely $\mathfrak h$" I mean that the stabilizer is $\exp\, \mathfrak h \subset \operatorname{Inn}$.) So the space of choices is a homogeneous space for $\operatorname{Inn}(\mathfrak g)$ (which is the smallest group integrating $\mathfrak g$) modeled on $\operatorname{Inn}(\mathfrak g)/\exp(\mathfrak h)$. But, anyway, the point is that $\exp(\mathfrak h)$ acts nontrivially on $\mathfrak g$ but, as I've said, trivially on the Dynkin diagram, and hence trivially on the group that you construct from the Dynkin diagram.
This might be why you don't like the notion of root systems: you really do need to make choices to identify algebras with their root systems. It's something like picking a basis for a vector space — great for computations, but not very geometric. As a precise example: for $\mathfrak{sl}(V) = \{x\in \operatorname{End}(V) \text{ s.t. }\operatorname{tr}(x)=0\}$, a choice of root system is the same as a choice of (ordered) basis for the vector space $V$.
On the other hand, I claim that you should like the representation theory of a LAG. One way to study this representation theory (I might go so far as to say "the best way") is to pick a root system for your LAG and look at how $\mathfrak h$ acts, etc. Then, for example, finite-dimensional irreducible representations of a semisimple Lie algebra $\mathfrak g$ correspond bijectively with ways to label the Dynkin diagram with nonnegative integers. So you can really get your hands on the representation theory.
But representations of a group $G$ is a very geometric thing. There is some sort of "space", called "$BG$" or "$\{\text{pt}\}/G$", for which the representations of $G$ are the same as vector bundles on this space. If you don't like thinking about "one $G$th of a point", there are homotopy-theoretic models of $BG$.
You should also think of categories as geometric. Think about the case when $G$ is a (finite) abelian group. Then you might remember Pontrjagin duality: the irreducible representations of $G$ are the same as points in the dual group $G^\vee$. Then, at least for $G$ a semisimple LAG, you might think of its finite-dimensional representations as being like the points of some "space" $G^\vee$. The difference is that in the abelian case, all the points on $G^\vee$ correspond to one-dimensional representations, whereas in the semisimple nonabelian case in general the points are "bigger". The points are parameterized by the positive weight lattice, but they aren't actually the positive weight lattice. But the space "$G^\vee$" is some space noncanonically-isomorphic to the positive weight lattice. Again, this is like how the Euclidean plane is noncanonically isomorphic to the Cartesian plane.
Let me explain how both kinds of foldings of Dynkin diagrams (i.e., $A_{2n-1} \to B_n$ and $A_{2n-1} \to C_n$) arise in the context of Lie algebras and characters of their representations.
First of all, what I will call the "standard combinatorial procedure" for folding a root system $\Phi$ according to a Dynkin diagram automorphism $\sigma$, as described by Stembridge here, will produce the Type $B_n$ diagram from a Type $A_{2n-1}$ diagram (and will produce a Type $C_n$ diagram from a Type $D_{n+1}$ diagram). The standard procedure is to produce the root system whose simple roots $\beta_{I}$ correspond to orbits $I \subseteq \Delta$ of the simple roots of the original diagram under the automorphism $\sigma$: we just take the sum of the roots in each orbit $\beta_{I} := \sum_{\alpha\in I} \alpha$. Let me call this folded diagram root system $\Phi^{\sigma}$.
However, if $\mathfrak{g}$ is the Lie algebra of $\Phi$, then the automorphism $\sigma$ acts on $\mathfrak{g}$ in an obvious way, and the fixed-point Lie subalgebra $\mathfrak{g}^{\sigma}$ has as its root system the dual of $\Phi^{\sigma}$. In this way we get the inclusions $\mathfrak{sp}_{2n}\subseteq \mathfrak{sl}_{2n}$, and $\mathfrak{so}_{2n} \subseteq \mathfrak{so}_{2n+1}$ (i.e. $A_{2n-1} \to C_n$ and $D_{n+1} \to B_n$).
But there is way to make the "standard" folded root system $\Phi^{\sigma}$ appear in the context of Lie algebras as well, namely, by considering so-called "twining characters." Let me call the Lie algebra associated to $\Phi^{\sigma}$ the "orbit Lie algebra" of $(\Phi,\sigma)$.
The set-up in which the orbit Lie algebra arises is this: we can "twist" any representation $V$ of $\mathfrak{g}$ by the automorphism $\sigma$ to get a new, twisted representation $V^{\sigma}$; if $V=V^{\lambda}$ is the highest-weight representation with highest weight $\lambda$, then $V^{\sigma} = V^{\sigma(\lambda)}$, where $\sigma$ acts on the weight lattice of $\Phi$ in the obvious way. Suppose that we choose a $\sigma$-fixed weight $\lambda$. Then we can view $\sigma$ as a map $\sigma\colon V^{\lambda}\to V^{\lambda}$ (I think this is technically defined up to scalar). The twining character of $V^{\lambda}$ is defined to be $\mathrm{ch}^{\sigma}(V^{\lambda})(h) = \mathrm{tr}(\sigma\cdot e^{h})$ for $h \in \mathfrak{h}$, just like the usual character would be $\mathrm{ch}(V^{\lambda})(h) = \mathrm{tr}(e^{h})$. The twining character formula, which is originally due to Jantzen (see the discussion at the beginning of https://arxiv.org/abs/1404.4098) but has been rediscovered by many people (e.g., https://arxiv.org/abs/hep-th/9612060, https://arxiv.org/abs/q-alg/9605046), asserts that the twining character $\mathrm{ch}^{\sigma}(V^{\lambda})$ is equal to the usual character $\mathrm{ch}(U^{\lambda})$ where $U^{\lambda}$ is the highest-weight representation of the orbit Lie algebra with highest weight $\lambda$ (note that since $\lambda$ is fixed by $\sigma$, it is naturally a weight of the folded root system $\Phi^{\sigma}$). So the upshot is that a Type $A_{2n-1}$ twining character is a Type $B_{n}$ ordinary character.
The twining characters have some interesting applications to combinatorics when considering "symmetric" versions of combinatorial objects associated to Lie algebras, which is how I became aware of them. I quote from the 2nd column of the 4th page of this paper of Kuperberg (https://arxiv.org/abs/math/9411239):
As stated in the proof, $\sigma_B$ is a Dynkin diagram automorphism. The character theory of semi-direct products arising from Dynkin diagram automorphisms is described by Neil Chriss [2], who explained to the author that although this theory is known to several representation theorists, it may not have been previously published. The group $\mathbb{Z}/2 \ltimes_{\sigma_B} SL(2a)$ has two components. The character of a representation on the identity component is just the usual character of $SL(2a)$. The character on the $\sigma_B$ component, when non-zero, equals the character of an associated representation of the dual Lie group, in this case $SO(2a + 1)$, to the subgroup fixed by the outer automorphism, in this case $Sp(2a)$. The representation associated to $V_{SL(2a)}(c\lambda_a)$ is the projective representation $V_{SO(2a+1)}(c\lambda_a)$, where $\lambda_a$ is now the weight corresponding to the short root of $B_a$, the root system of $SO(2a + 1)$. In particular, the trace of $\sigma_B$ is the dimension of $V_{SO(2a+1)}(c\lambda_a)$, as given by the Weyl dimension formula, and the trace of $\sigma_BD_q$ is the q-dimension, as given by the Weyl q-dimension formula.
The disclaimer at the beginning of this quote suggests that (at least in 1994) this folding business was not well-known or written down precisely in a canonical text.
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Homology and homotopy groups