Fix broken links and check in CI that we don't introduce new ones
I had to change the linkification of the local paths in the README.md.
CC @nickm
Fixes #238 (closed) (at least, the part I think I want to do now).
Assuming we merge both this and !193 (merged), I'll make a followup to apply the changes from !193 (merged) to the new CI job added here.