I just installed the Soul Reaver patch that was released today (9/13/2023), and in the patch notes it says an Xbox Series X controller should be fully supported now, but mine doesn't seem to work outside of the main menu. Any clues about what's going on?

Also my saves seem to have disappeared.

Go to the options (either from the main menu or by pressing Esc in game), move down to "Control Keyboard", and tick it over to "Control Gamepad". From there, the default controls don't match the original. It would be nice to see them patched, but it's pretty simple to tweak.

UPD: GOG didn't actually say Full Xbox support. They say full gamepads support. Next Xinput gamepads listed as supported and they now indeed supported. It didn't say "incl full suppor for". Previously they not worked at all. Even Dinput ones worked 50/50 for pads come after 2000 year. Puting words full gamepad support and put a list there are both correct changelog from GOG.

It is wrapper for old game. Not a native support. And that mean there is no such thing as default control schema for gaming input devices (also avoid hotplug/unplug devices and few else legacy limitations/optional features).

Just like for previous version. And any controller.
So it's not real "full support for Xbox/etc". Today that mean auto bind too. GOG miss it. It is mproved but not like native nowadays (from '06+) Xinput game with native Xbox pad.
It is random for me for old version how old version did work with analog pads. But there is at least 50/50 chances to issue. Should be solved now. Shouldn't be "going in circles" issue but nothing extra.

So for every pre-'06 game go to controls reset to default and rebind manually with gamepad. Specific support always be optional.
For games after only Xbox/clone not needed to bind. Sony, else, General pads still should be rebinded if game allow. Now only Xbox is mandatory support else are optional again. And there is very rare exceptions when game did FULL support specific pads.
But not with wrapper. It's same as connect racing wheel kit. Game have no idea what a device you have to correctly represent buttons layout.
And GOG shouldn't use "full support" phrase cause it's still working not like nowadays games. But that my IMO. Full support is like connect Xbox and get Xbox buttons layout, automatically properly set. And if it's say Nintendo full support - same automatic preset and button promts for Nintendo. Not just compatability.

Just choose gamepad (instead of keyboard) in the game menu options.

Your old saves were just removed because gog added strange fix which prevents to use old saves in the game with this update (and new saves in the game without this update). Say thanks for this gog programmers with really bad hands.

For saves and if he use GOG Galaxy build its probably somehow related to Cloud Support. Like Steam first look to saves in own subfolder user data, Galaxy have account folder somewhere in AppData. Even some offline builds use this location for saves (account0 folder for offline build).

Wrace
no I bought it here when SR got delisted from series bundle. As I understand its not a big difference after BO1 wrapper (and exe from crack/fixes with game slowdown) and after private branch test build of SR1. Although it is called build from 07.2023 and build date also from that date.

http://www.gogdb.org/products?search=kain

I hope Allard can provide you info you looking for.
I can say only for other four games files. I can say for previous version of SR I download from P2P all used your Bink patch only English one have your credits in intro.
Now by files its looks like reverted to original 1.2 RealVideo format + ddraw and dinput wrappers from GOG
https://www.gogdb.org/product/1207659030/build/56678180399683452