In an algebraic topological setting you would use singular homology to define the Euler characteristic. Hence, you need homology groups to define it (in its most general setting). So I think it makes more sense to discuss it last.
You can of course also define the Euler characteristic in a weaker setting by just considering simplicial complexes. This then reduces to the setting of polyhedral surfaces. However, if you use this definition it becomes a bit more restrictive.
So I guess it makes more sense to discuss how homotopy and homology connect to the Euler characteristic then the other way round.
The Euler characteristic was originally defined without the use of homology and I give it as a simple example here because most people do not know what homology is. Of course it can be defined using the Betti numbers or ranks of the homology but this came later.
Then you should say "that the Euler characteristic gives us a well defined invariant of a polyhedral surface which is easy to compute" instead of "that the Euler characteristic gives us a well defined invariant of a topological space which is easy to compute" since for a topological space you require a more general version of the Euler characteristic.