[Feature request] Always save/keep custom gamepad key bindings after changing handler

[ElTioRata]

Quick summary

Follow up to #14773

Some quotes of mine from the issue:

I had a key mapping configured for a game, then I decided to try it with a controller today so I switched the handler from Keyboard to XInput but then when I was done and I changed the handler back to Keyboard I saw that it didn't save my previous key mapping.
That's terrible for the users since they'd expect the bindings to be saved, and if they aren't aware of that and don't backup the bindings before changing the handler, they'd have to re-configure everything again and that could take either a couple of minutes to an entire session of in-game mechanics testing to do re-configure everything again. I'm making a separate issue as a feature request since this is not an issue on most emulators.

[carlmylo]

Sounds similar to to this feature request I opened up earlier. It's absolutely frustrating for us rhythm game players since we usually swap controller types constantly, and I had a feeling this also extended past rhythm games as well,