In Fedora, we run the following check when we build Python documentation:

# Verify that all of the local links work
#
# (we can't check network links, as we shouldn't be making network connections
# within a build.  Also, don't bother checking the .txt source files; some
# contain example URLs, which don't work)
linkchecker \
  --ignore-url=^mailto: --ignore-url=^http --ignore-url=^ftp \
  --ignore-url=.txt\$ --no-warnings \
  Doc/build/html/index.html

From time to time, it discovers broken links:

  https://github.com/python/cpython/pull/15700
  https://github.com/python/cpython/pull/20383
  https://github.com/python/cpython/pull/20388

It would be really nice if this check run as part of the CI that builds the documentation.