I might have just the thing: a book called Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra by Jiri Matousek (information here: https://bookstore.ams.org/stml-53), which covers some 'well-known mathematical gems' as well as some lesser-known results, mostly in combinatorics and graph theory, and requiring only a modest amount of linear algebra as a prerequisite.
The book Linear Algebra in Action by Harry Dym is also worth a look. After the first dozen or so chapters covering the standard material, he outlines many applications in other (mostly analysis-related) areas of math, such as difference and differential equations, extremal problems and convex analysis, dynamical systems, Markov chains, and even some complex analysis!
As for LA books that cover computation of homology groups, for instance, I'm guessing these are rather rare, as this topic requires so much background material that expecting it to be covered in a linear algebra book (introductory or otherwise) is unrealistic. Also, if you're interested in graph theory applications in particular, just pick up any book on 'spectral graph theory', a field that analyses graphs by representing them with matrices and studying their spectra (i.e., sets of eigenvalues), and is unsurprisingly very linear-algebraic.
Lastly, and for the record, I do actually find the exercises in Axler to be super-exciting!