Several issues with version switcher

[drammock]

Currently the version switcher on main is broken, and there are several open (or recently closed) issues/PRs about it. This issue is an attempt to get everyone on the same page:

• fix: version switcher on sidebar (#1340) #1342 was a quick fix to the sidebar version switcher not working, which I think is what caused the non-sidebar switcher to break. It addressed the report in Version switcher not opening on mobile #1340. I think it's problem was not cloning the nodes before appending.
• fix: version selector JS #1305 is an older attempt that I think actually fixes Version switcher not opening on mobile #1340 (and Version switcher JS does not add version spans to both menus #1304 of which Version switcher not opening on mobile #1340 is a duplicate). It got mired down in trying to fix too many other things at once, which is why it's still not merged.
• Redirecting to homepage broken for version switcher #1343 reports other issues --- redirects not working correctly on some pages. This might be fixed by fix: version selector JS #1305 too but I haven't tested it yet.
• fix: resolve path of switcher when using relative #1344 tries to fix version switcher on local builds

IMO next step should be to move forward with #1305 as the starting point, but set aside the remaining issues with aria roles (and open a new issue to track that unfinished work). I'll try to get to that later today, but if others have time feel free to take a look and review / propose changes to what is currently there.

cc @12rambau @lagru @alphapapa @jpfeuffer

[12rambau]

I tried to work on #1343 but I don't manage to reproduce the bug locally. Even when I set up a server, the redirection is working as expected. So instead I'm focusing on #1344 first (so that PR build can be used as real life test material).

[drammock]

when I recently helped NumPy update to theme version 0.13.3 the version dropdown was working in PR builds and also on local builds (numpy/numpy#23647 (comment)) but I guess that's because their json_url is a real URL (not a relative path)?

[drammock]

closing in favor of #1357 and #1358. I'll add the block-release tag to #1357 instead of this one.