[Doc] Fix cached toctree behavior

[peytondmurray]

Why are these changes needed?

This PR fixes some issues with the sidebar that arose from caching the HTML for the sidebar. In short, I had issues getting the JS to store the checkbox state to work. I solved this in python, by reusing the cached toctree HTML but setting the "checked" state on the appropriate dropdown inputs on each page using beautifulsoup. The operation comes with an overhead, but it's still quite small compared to reconstructing a new toctree every page write.

Related issue number

Partially addressed ray-project#37944.
Targets ray-project#41115.

Checks

• I've signed off every commit(by using the -s flag, i.e., git commit -s) in this PR.
• I've run scripts/format.sh to lint the changes in this PR.
• I've included any doc changes needed for https://docs.ray.io/en/master/.
  • I've added any new APIs to the API Reference. For example, if I added a
    method in Tune, I've added it in doc/source/tune/api/ under the
    corresponding .rst file.
• I've made sure the tests are passing. Note that there might be a few flaky tests, see the recent failures at https://flakey-tests.ray.io/
• Testing Strategy
  • Unit tests
  • Release tests
  • This PR is not tested :(