There was no difference, including with different input methods instead of just xInput. I tried these controllers while being connected to the RPI with USB instead of BT. I updated the firmware of the gamepads to the latest.

I'm trying to use the xInput method because I've read that is the most compatible, but I tried the others just to see if it would make a difference. I've got two SN30 gamepads from 8bitdo and the issue occurs with both gamepads under all four input methods (although I think under at least one of the methods, it doesnt work at all - not even in the SteamLink menu). Especially when the games work fine with the official button templates when using a Windows-based Steam host. But obviously, mapping keys for every game I have is time consuming and not something I want to do. If I do manually map keys using a new template, the games I'm having problems with work fine.

It is also recognized by the test input function in the controller section of Settings while connected to the Linux host via SteamLink. My controller is fully recognized by the SteamLink launcher when testing and mapping keys. This is not necessary when playing locally on my Linux host (no Steam Link) with the same gamepad all of these games (and others) work with the official button templates, without customization in most cases.

With Steam Link via the Linux host, the controller works within the big picture mode menus, but not under the games - UNLESS I manually map keyboard keys or mouse buttons to the gamepad buttons. I've been trying each test scenario with five different games that are supposed to have full controller support (Cuphead, Stardew Valley, Lara Croft Go, Kingdom, Wolfenstein II New Colossus) just for consistency, but I've tried other games at times as well. I've tried many other settings, devices, and combinations, to see if I could find anything that does work. My goal is to have the SN30 gamepad working with my Linux host via SteamLink on the Raspberry Pi (because I prefer Linux), connected via Bluetooth, using the xInput method. Skipping ahead a bit - everything works perfectly from my RPI and Chromebook if I instead boot the same PC into Windows 10 and host Steam games from there. But when I use the Steam Link app to connect to the same Linux host from either my Raspberry Pi (3b), my HP Chromebook x2 (ARM based), or my Samsung TV, the same controller will no longer work with these games (or any others that I've tried). I've got an 8bitdo SN30 controller that works fine with many Steam games under xInput mode when connected via Bluetooth directly to my Linux host (Linux Mint 20.3).

8bitdo pro 2 steam