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The previous msi installer was working fine with `wine msiexec /i python*.msi`, but the new exe-based one fails with an unreadable error in latest wine.

The latest wine may not be up to date enough, since the new installer (and new Python) depend on MSVC 14.0, which is still only in preview.
Of course, there could be other issues, but without any more information it's very difficult to act on them.

Maybe you could continue to distribute the msi? I’ve also heard it’s used by Windows admins to install something on many computers at once.

The new installer can also be used for that, and it's actually documented now (at least in the CHM - doesn't seem to have made it onto the website yet). I personally install it onto many computers at once fairly often, so I'm going to make sure that works.
The old single MSI installer is completely gone though and won't be coming back.

I just tried it with wine-git, it doesn’t go further either by running the exe directly, or by running msiexec.exe /i on it.
Is there anything else required to run it?

You should just run it directly - it isn't an MSI.
As I said, wine is probably not compatible with the new CRT version yet, but without at least an error message, it's impossible to tell.

The installer runs, but the first screen doesn’t display any of its buttons (I can still click on them).
After trying to start the actual installation, I get "0x80070005 - Access Denied" (despite having the permissions on my wine folder, of course). Whether I select system-wide or local installation doesn’t change anything.
I attached the logs wine outputs on stderr, not that they seem very useful.

Yeah, I don't get anything helpful out of that, except that it seems wine hasn't implemented some of the APIs that it needs to run the installer. You may get a more helpful response from one of their devs, but I'm well out of my depth sorry.

There doesn't seem to be information here that can help do anything about it, and version 3.5 is no longer maintained.
Please create a new issue if you are still having this problem with current versions (>= 3.9).