One way to approach this question quantitatively is suggested by probability. One can put various measures on the space of all simplicial complexes on $n$ vertices. One perhaps fairly natural measure is to take a random graph and then take the clique complex. This doesn't give us all complexes on $n$ vertices but every complex is homeomorphic to the clique complex of some graph, so we are covering everything up to homeomorphism as $n \to \infty$.

The main point of my paper Topology of random clique complexes is that almost all simplicial complexes arising this way are fairly simple topologically. In particular is shown that for a typical $d$-dimensional clique complex, the homology groups $H_k$ all vanish when $k > \lfloor d/2 \rfloor$ and when $k< d/4$, and that almost all of whatever homology remains is concentrated in the middle dimension $k=\lfloor d/2 \rfloor$.

It is currently an open problem to decide whether the homology is vanishing (or merely small) between $k=d/4$ and $k=d/2$. If one could establish this, then one would be well on the way to showing that almost all flag complexes are homotopy to a wedge of spheres; indeed the last thing to do would be to rule out torsion in middle homology with integer coefficients.

I don't have a good feel for whether either of these things is even true, but I do think that this paper gives good anecdotal evidence that most flag complexes are somewhat simple topologically, and is a step in the direction of answering Forman's question. (This particular measure seems especially natural from the point of view of combinatorics, since so many simplicial complexes arise as order complexes of posets, hence are automatically flag complexes.)

UPDATE:

(1) I showed recently that for every $k \ge 3$, there is a range of edge probability so that the random clique complex (also called random flag complex) is rationally homotopy equivalent to a wedge of $k$-dimensional spheres. In particular all the rational homology is in middle degree. There is only a very small overlap where there is homology in degree $k$ and in degree $k+1$, but in some sense, most of the time there is only homology in one degree. The conjecture that "rationally homotopy equivalent" can be replaced by "homotopy equivalent" is equivalent to showing that with high probability, homology is torsion free.

(2) On the note of torsion in random homology, in joint work with Hoffman and Paquette, we recently showed that for a slightly different model of random simplicial complex, for most of the range where rational homology is vanishing, integer homology is also vanishing.

There are one or two technical issues in applying the method of (2) in the setting of (1) (namely non-monotonicity of homology), but so far it seems like there is reason to believe that the method will go through eventually.

Together, these two recent results suggest that a random flag complex (for a suitable range of edge probability $p$) is homotopy equivalent to wedges of $d$-dimensional spheres. Random flag complexes seem to me like a very natural model for addressing your question probabilistically, since so many complexes in combinatorics are flag complexes, arising as order complexes of posets, etc.