We will prove this by induction: suppose, we know that if $X$ consists of $\,<n$ cells then for all other finite $CW$-complexes $X'$ such that $X\approx X'$ we have $\chi(X)=\chi(X')$.
Let $Y$ and $Y'$ be two finite $CW$-complexes, $Y$ consists of $n$ cells, and $f:Y\to Y'$ is homotopy equivalence. Consider one higher-dimensional cell of $Y$: let $Y=Z\cup_\alpha D^k$, where $Z$ consists of $n-1$ cells, $D^k$ is just a cell and $\alpha:\partial D^k\to Z$ is attaching map. We see that $\chi(Z)=\chi(Y)-(-1^k)$.
Then consider the space $CZ\cup_{f|_Z}Y'$, here $CZ$ is a cone. $f$ supposed to be cellular map, so $CZ\cup_{f|_Z}Y'$ is a $CW$ complex; it contains all cells of $Y'$, all cells of $Z$ times $I$, and the vertex of the cone, thus $\chi(CZ\cup_{f|_Z}Y')=\chi(Y')-\chi(Z)+1$. But we know that this space has homotopy type of $S^k$, so by inductive hypothesis $\chi(CZ\cup_{f|_Z}Y')=1+(-1^k)$, and $\chi(Y)=\chi(Y')$ as desired.
EDIT: for the induction we also need the statement be truth in case $X\approx pt$ and $X\approx S^m$. If $X\approx S^m$, we may glue an $m+1$-disc and obtain contractible space.
Now suppose $X\approx pt$, $m$ is maximal dimension of cells of $X$ and $X$ has $p$ $\,m$-cells. The space $sk_{m-1}(X)$ is $(m-2)$-connected, so it is homotopy equivalent to the bouquet of $q$ $\,(n-1)$-spheres. Gluing $m$-cells determines a homomorphism $\phi:\mathbb Z^p\to\mathbb Z^q$, and equalities $\mathrm{coim\,}\phi=\pi_{m-1}(X)$ and $\ker\phi\subseteq\pi_m(X)$ give us $p=q$. Then we may remove $p$ $\,m$-cells, $p$ $\,(m-1)$-cells, and repeat. (this reasoning sounds more like homological argument)
Here is an element of answer, which is by no means complete, but it provides at least a first motivation for abstract homotopy theory. I assume that by "abstract homotopy theory" you mean the study of Quillen model categories.
Daniel Quillen realized that the basic machinery of homotopy theory of topological spaces can be set up in the more general context of categories equipped with three classes of morphisms satisfying a few axioms reminiscent of properties of topological spaces, the so-called Quillen model categories.
This situation has some advantages. First, it gives you an abstract approach and a unique language to talk about homotopy theory in a large number of different settings. Some of them are geometric in nature, like spaces, diagrams of spaces, spectra, while others are not, like chain complexes and simplicial sets. To the delight of a topologist, now a question like "What is the suspension of an augmented commutative algebra" makes sense. Second, all these settings have their computational peculiarities, and one of them, at least in some context, can reveal itself more convenient.
Moreover, the category of spaces is not particularly nice and it lacks some good categorical properties, indeed it is not cartesian closed or locally cartesian closed. To get a cartesian closed category you need to restrict to the subcategory of compactly generated spaces, but this category is not locally cartesian closed. In order to get this last property you can use a more combinatorial/algebraic setting instead, like the topos of simplicial sets.
Finally, it is possible to capture in a specific sense, through the notion of Quillen equivalence, what it means for two homotopy theories to be the same. In particular, there exist a model category on topological spaces and a model category on simplicial sets that are Quillen equivalent.
Best Answer
I have not read Hatcher, but Prism's go back to Eilenberg-Steenrod. The idea is simple if $H(x,t)$ is a homotopy on $\Delta_p$ (just assume for the moment it is constant on the boundary), then we can view $H$ as a function on the prism $I\times \Delta_p$. Now the boundary of the prism can be thought of as the difference of the top and the bottom, (the sides are constant). This can be then extended to sums of simplices. And so it shows that homotopy gives a chain-homotopy which induces and isomorphism.
I think the problem arises because $I\times \Delta_p$ is no longer a simplex. Athough personally I have never really understood why this is such a big issue since one can easlily and explicitly write a prism as a sum of simplices.