It's not really complicated, it's just a little difficult to find the right files you need.

> Running from the Add Remove Programs entry prompts for admin access.

Which clearly shows it was previously installed by an admin, and so you'll need that to remove/repair/upgrade it properly. If you aren't an administrator on your machine, you'll need to find out who is and ask them to do it.

> Directly running "repair" in the rc2 installer with admin says "No Python 3.10 installation was detected"

This generally means you need a different copy of the installer. There should only be 32-bit and 64-bit ones floating around, but it may be that you actually need RC1 and not RC2? They are all subtly different (most notably in their primary UUID ;) ) and won't repair each other (which would be an upgrade/downgrade, which should work when you haven't deleted all the files manually).

Otherwise, there used to be tools out there that could delete all the registrations in the Windows installer database, but I haven't had to use one in years and I have no idea which would work well. If you can remove all the registration entries (which you can't do by hand, trust me, you need a clever tool) then you should be able to install a new version without hitting these issues.