Add automated testing for examples in README

[elpdt852]

In order to make sure the examples in README is well maintained, we should add automated testing to the examples and installation instructions. There are two approaches we can take:

1. Generate README with examples in individual nix files that we test
2. Extract examples out of README for testing

I think it's probably cleaner to do (2), so that it's easier to maintain the README with no generation step. Let's find out if what solutions there are for (2) and if any nix project does something similar.

For testing against home-manager, we likely want a development flake, as we don't want to introduce home-manager as a top-level dependency.