This may not be a helpful answer to the precise question, but in the 1970s I got disillusioned with homology as a first course in algebraic topology and started teaching knot theory; the examples are immediate, and one gets links with group theory, and with some homological type ideas, via the free differential calculus. Students could quickly see what the subject was about, and it showed that you were getting some answers but incomplete ones. Also there are lots of nice computational examples to do.

It also fits with my view developed over the years that algebraic topology should be and can be more nonabelian, to reflect the geometry. For example, one would really like to write the boundary of the standard diagram of the Klein Bottle as $a+b-a+b$, not just $2b$.

This knotty activity led to me giving general lectures on "How mathematics gets into knots", and into all sorts of things, including making a mathematical exhibition and promoting the work of the sculptor John Robinson.

Others took over the course and even got a nice book out of it ("Knots and surfaces" by Gilbert and Porter).