3.7 is probably broken because it used an older version of the theme, and the switcher json may have changed a bit since.

The rest are broken because our version_match setting is wrong; it currently picks either 'stable' or 'devdocs', neither of which are used in the switcher.json. Prior to #28512, 'stable' was a thing, but this broke the banner so it was changed to the actual version. And it makes no sense to use 'stable'; as you have noted, once a version becomes older, it's no longer 'stable'. And devdocs actually have been set to a version of 'dev' since #23949.

• Start at: https://matplotlib.org/stable/ - the switcher shows "choose version" (?)

I pushed matplotlib/matplotlib.github.com@4eabb2b which applies the linked PR to the published docs, and fixed these pages.

• In the switcher select "3.10 (dev)" - you are directed to the correct page but the switcher still shows "choose version"

This should be fixed by #28644.

• In the switcher select "3.9 (stabe)" - you are directed to the correct page but the switcher still shows "choose version"

See first point.

• In the switcher select "3.8" - you are directed to the correct page but the switcher still shows "stable"

I pushed matplotlib/matplotlib.github.com@fd6b5d2 which backported the theme fixes and fixes these pages.

• In the switcher select "3.7" - you are directed to the correct page but the switcher again shows "stable" and is now broken (i.e. does not have a dropdown anymore)

Here the JavaScript raises on missing jQuery before it can populate everything; possibly we updated Sphinx and not pydata-sphinx-theme, but I'm not sure exactly which versions were used for that build. I pushed matplotlib/matplotlib.github.com@1fca7e7 which restores jQuery on the files, and fixes populating the dropdown.

And then I also backported pydata/pydata-sphinx-theme@59401a6#diff-dff36566ae5c858672e0d5a02ec6a57756f11ff11751e0cf7509098fb0941234R308 via matplotlib/matplotlib.github.com@691be21 which fixed 3.6 and 3.7

Here the JavaScript raises on missing jQuery before it can populate everything; possibly we updated Sphinx and not pydata-sphinx-theme, but I'm not sure exactly which versions were used for that build. I pushed matplotlib/matplotlib.github.com@1fca7e7 which restores jQuery on the files, and fixes populating the dropdown.

One more followup here which fixes the jQuery references for files 3.7 that are in subdirectories: matplotlib/matplotlib.github.com@8fd7442