There are maps $|Sing(X)| \to |\underline{Sing}(X)| \to X$ which realize to weak homotopy equivalences. The inclusion of the n-skeleton $|Sing(X)|^{(n)}| \to |Sing(X)|$ is n-connected, because this is always true for CW-complexes, and so the map $|Sing(X)|^{(n)} \to X$ is n-connected. You can't really do any better than this estimate because the n-skeleton has zero homology groups in degrees above n.
The simplicial space $\underline{Sing}(X)$ contains the sub-simplicial space of constant simplices $\Delta^n \to X$. This is homeomorphic to $X$ itself and so, if we write $cX$ for the constant simplicial space with value $X$, we get a map $cX \to \underline{Sing}(X)$. This inclusion $X \to \underline{map}(\Delta^n,X)$ is a homotopy equivalence because the simplex is contractible, so this map of simplicial spaces is levelwise a weak equivalence. The geometric realization of $cX$ is $X$ itself, and so is its n-skeleton for all n. An excision argument (which takes some work) will show that the same is true for the simplicial space (at least under good conditions), and so each of the skeleta of $|\underline{Sing}(X)|$ is homotopy equivalent to $X$.
Therefore the map on n-skeleta is n-connected. I realize that this is the "compare with $X$" game that you mentioned, but my point is that because the simplicial space $\underline{Sing}(X)$ is homotopically constant the comparison $|Sing(X)| \to |\underline{Sing}(X)|$ really is comparing with $X$. From the point of view of homotopy theory it's not arising from good levelwise structure of the map at all, but comes from the simplicial assemblage.
I don't know whether this helps your generalization.
I'm not aware that the model category you want has been constructed. But it seems like an interesting question. You should ask Julie Bergner if she has thought of anything along these lines.
I don't have an answer, but I will think out loud for a bit.
(I'll be a little vague by what I mean by "space", but I probably mean "simplicial sets" here.)
I will assume that your property 3 should say: "The weak equivalences between fibrant objects (i.e., between Segal spaces) are the DK-equivalences." I would also like to throw in an additional property:
4. The trivial fibrations between Segal spaces are maps $f:X\to Y$ which are DK-equivalences, Reedy fibrations, and such that the induced map $f_0:X_0\to Y_0$ on $0$-spaces is surjective. (Note: since $f$ is a Reedy fibration, $f_0$ is a fibration of spaces.)
Yes, this comes out of thin air ... but it's modelled on the trivial fibrations in the "folk model structure" on Cat. You could go further, and posit that fibrations between Segal spaces are Reedy fibrations such that $f_0$ is surjective.
Given a space $U$, let $cU$ denote the "$0$-coskeleton" simplical space, with $(cU)_n=U^{\times (n+1)}$. If $U$ is a fibrant space, then $cU$ is Reedy fibrant; if $U\to V$ is a fibration, $cU\to cV$ is a Reedy fibration. Furthermore, $cU$ clearly satisfies the Segal condition.
Thus, if $g:U\to V$ is a surjective fibration of spaces, $cg: cU\to cV$ should be a trivial fibration in our model category, according to 4.
The functor $c$ is right adjoint to $X\mapsto X_0$: that is, maps of simplicial spaces $X\to cU$ are naturally the same as maps $X_0\to U$ of spaces.
Putting all this together, we discover that, if such a model category exists, a cofibration $f: A\to B$ should have the following properties: the map $f_0 : A_0\to B_0$ is a cofibration of spaces, and $B_0=B_0'\amalg B_0''$ so that $f_0$ restricts to a weak equivalence $A_0\to B_0'$, and such that $B_0''$ is homotopy discrete (i.e., has the weak homotopy type of a discrete space).
In particular, a necessary condition for $B$ to be cofibrant is that $B_0$ is homotopy discrete.
This is a pretty restrictive condition on cofibrations, but it does not seem impossible. If there actually was a model category with all these properties, it appears that the class of fibrant-and-cofibrant objects would be what you might call the quasi Segal categories. These are the Segal spaces $X$ such that $X_0$ is homotopy discrete. Cofibrant replacement of a Segal category would give a DK-equivalent quasi-Segal category.
That would be a pleasing outcome, and probably along the lines of what you're looking for.
Best Answer
If $\mathcal{A}$ is an abelian category, then the Dold-Kan correspondence supplies an equivalence between the category of simplicial objects of $\mathcal{A}$ and the category of nonnegatively graded chain complexes in $\mathcal{A}$. One can therefore think of simplicial objects as a generalization of chain complexes to non-abelian settings.
In homological algebra, chain complexes often arise by choosing ``resolutions'' of objects $X \in \mathcal{A}$: that is, chain complexes $$ \cdots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0$$ with homology $$H_{i}( P_{\ast} ) = \begin{cases} X & \text{ if } i = 0 \\ 0 & \text{ otherwise }. \end{cases}$$ Among these, a special role is played by projective resolutions: that is, resolutions where each $P_n$ is a projective object of $\mathcal{A}$.
If $X_{\ast}$ is a simplicial space, it might be helpful to think of $X_{\ast}$ as a resolution of the geometric realization $| X_{\ast} |$. It plays the role of a ``projective resolution'' if $X_{\ast}$ is degreewise discrete: that is, if it is a simplicial set.