I think your intuition of founding algebraic topology through homotopical methods is sound, and I am pleased to report that this has been done in the two papers:

R. Brown and P.J. Higgins, ``On the algebra of cubes'', *J. Pure
Appl. Algebra* 21 (1981) 233-260.

R. Brown and P.J. Higgins, ``Colimit theorems for relative homotopy
groups'', *J. Pure Appl. Algebra* 22 (1981) 11-41.

In the space of less than 60 pages, and *without using simplicial approximation* or *singular homology*, these papers prove

the Brouwer Degree Theorem (the $n$-sphere $S^n$ is
$(n-1)$-connected and the homotopy classes of maps of $S^n$ to
itself are classified by an integer called the *degree* of the
map);

the Relative Hurewicz Theorem, which is seen here as describing the
morphism $$\pi_n(X,A,x) \to \pi_n(X \cup CA,CA,x) \cong
\pi_n(X \cup CA,x)$$ when $(X,A)$ is $(n-1)$-connected, and so does
not require the usual involvement of homology groups;

Whitehead's theorem (1949) that $ \pi_2(X \cup \{e^2_{\lambda}
\},X,x)$ is a free crossed $\pi_1(X,x)$-module.

The last theorem allows, for the usual identification of a square $\sigma$ to give the Klein bottle, to write the nonabelian formula:

$$ \delta \sigma = a+b -a +b.$$
Whitehead's theorem is sometimes mentioned in texts but rarely proved. For more on the background to the theorem, see also http://ncatlab.org/nlab/show/free+crossed+module.

Actually one has here a method for direct calculation of some *homotopy $2$-types*, as crossed modules.

Historical background and intuitions for this method are given in an expository paper showing the origins of the idea of *cubical higher homotopy groupoid*. Not surprisingly, the method needs a number of new ideas, and departures from tradition, particularly the generalisation to higher dimensions of the fundamental groupoid and the Seifert-van Kampen Theorem.

The method is not to start with just a topological space but with a filtered space: i.e. a space $X$ and an increasing sequence of subspaces $X_n, n \geqslant 0$. This enables one to define a *crossed complex* $\Pi X_*$ using the fundamental groupoid of $X_1$ on the set $X_0$ of base points and for $n \geqslant 2$ the relative homotopy groups $\pi_n(X_n,X_{n-1},x), x \in X_0$, with the operations of $\pi_1$ and the standard boundary operations. The functor $\Pi$ satisfies a form of the Seifert-van Kampen Theorem, i.e. it can be computed on certain pushouts (and more generally) of filtered spaces. This gives the above results, and more. So one get new nonabelian calculations of second relative homotopy groups; and of higher relative homotopy groups as modules over a fundamental group, without using covering spaces.

However this result is not proved directly, but via a related (and not so trivial to establish!) construction of a *cubical higher homotopy groupoid* $\rho X_*$ of a filtered space. The first paper above establishes many key algebraic properties of these gadgets. These higher homotopy groupoids are *strict* structures, like the relative homotopy groups, and so a colimit type theorem allows for precise calculations.

The use of filtered spaces may be a surprise, though they are very common. However Grothendieck in Section 5 of "Esquisse d'un programme" argues that more structured concepts than a topological space are needed for the purposes of geometry: he likes stratifications. We use filtered spaces because in that context we can make these strict higher homotopy groupoids work, and express intuitions such as "algebraic inverses to subdivisions", for which simplicial or globular theories are not so well suited.

This work has been developed since those papers were published and some details of exposition modified: a long account is published as

R. Brown, P.J. Higgins, R. Sivera, *Nonabelian algebraic
topology: filtered spaces, crossed complexes, cubical homotopy
groupoids*, EMS Tracts in Mathematics Vol. 15, 703 pages. (August
2011).

(more details on http://pages.bangor.ac.uk/~mas010/nonab-a-t.html, including a downloadable pdf).

The above papers and most of the book have been written to be checkable by a graduate student with some knowledge of general topology and the fundamental groupoid, such as in my Topology and groupoids.

I have to give this plug for my work with Philip (and others) since the notion of a Higher Homotopy Seifert-van Kampen Theorem is to my knowledge not mentioned in any text in algebraic topology (except mine).

I have no idea how to do an analogue of this work for a filtered topos!