I've heard it said that the reason why the homology groups of a space are a computable invariant is because they are a stable invariant in the sense that they are stable under suspension.

I'm familiar with the standard computation which shows that for reduced homology ${\tilde H_n}(S X)\simeq {\tilde H_{n-1}}(X)$ for all $n$, where $S X$ denotes the suspension of a topological space $X$; just let $A$ be $S X$ with the top vertex removed and $B$ be $S X$ with the bottom vertex removed, and apply Mayer-Vietoris, noting that $A$ and $B$ are contractible, and that $A \cap B$ is homotopy equivalent to $X$.

Is this computation the sense in which my first sentence should be understood? That it enables us to compute the homology of certain spaces (such as spheres) which are built out of each other by suspension, by some kind of inductive argument? But other than the obvious example of spheres, are there many other examples of spaces whose homology can be computed by taking suspensions in this way? Are there other reasons why this "stability" of homology groups under suspension makes it much more computable than other invariants?