Question

This isn't quite what you mean, but I took Igor Frenkel's algebraic topology course as an undergrad. He taught out of Massey's book, A Basic Course in Algebraic Topology. It starts with the classification of 2-manifolds, does the fundamental group and the Seifert-von Kampen theorem, and then does singular homology and cohomology. De Rham cohomology is only there as an appendix. I think the fundamental group is a little bit easier to grasp early on in a first course than singular homology. For cohomology first, you could do something like Bott & Tu, I suppose, but I think this way is a bit more useful because de Rham cohomology is a little too nice for its own good.