If you are having problems getting your xbox controller to be recognized properly (like it happened to me) from patch 2 onward, you can solve the issue by removing steam_api64.dll from the installation folder. Be careful though, as this changes the save file location (it's still in "AppData\Roaming\TRX", but it doesn't use a sub-folder anymore), so you might want to copy your saves and the .dll before trying, just in case (and then copy the savefiles to the new location, so you don't lose your progress).
Alternatively, you could use GOG Galaxy to rollback to patch 1, but that's obviously not ideal.

I made GOG aware of the issue (although I don't know how many people are affected), so hopefully a proper fix will be implemented, eventually.

You saved my life, I was wondering why I couldn't remap action and jump without the game freaking out.

Couldn't get the weapon hot swapping to work on the Xbox One S controller (i.e. Change View BTN + B for shotgun) but after renaming "steam_api64.dll" to "steam_api64.dll.bak" everything work as expected.

Thanks a million!

Problem solved and thank you sir. In my case, the controller button icons were not appearing in the game while some buttons could be mapped and others could not. Your fix remedied this. The file "steam_api64.dll" must be what Steam uses for it's controller configuration. When I was trying to fix this problem, I plugged my controller into a friend's Steam version and it worked but if the Steam Input was disabled, the same situation occurred as with GOG. This file should be removed for GOG many thanks!!

Thank You so much @mdqp

I thought I was the only one with this issue.
I've spent so much time trying to fix/repair installation, rolling back to earlier versions.
Made a ticket at Aspyr support as well.

And it was such a simple solution.

Glad it was helpful, everyone. Someone kindly helped me over at the Tomb Raider Forums, and I figured I'd share the fix here as well. I am not sure what the .dll is for, exactly (I wonder if it might be used for other stuff, like the achievements on Galaxy), but the issues with rebinding keys and not getting the rumble effect anymore was annoying me too much, I couldn't let it go.

I hope this gets fixed properly one way or the other. There should be another patch for the game, so hopefully that's the chance for them to fix it perfectly.