Update Doxygen docs

Context

Updating all of our documentation leading up to the handoff next week.

Summary

(thanks @jroiseaux for finding some of these issues)

Made a handful of small changes:

• Updated DOCUMENTATION.md and fixed examples
• Configured Doxygen to ignore the legacy code
• Fixed the make docs job to not fail if the public/ directory already exists
• Removed some links at the bottom of README.md -- these don't work consistently between the static site and the GitLab markdown view, plus the files we were linking to there are easily discoverable anyway.

How was this tested?

Generated the docs locally to make sure that it's not picking up the code under code/old/.