With Steam Input enabled, games don't detect Xbox Controller

**Game information**

Hollow Knight, Fatal Fury City of the Wolves, Is This Seat Taken, Absolum Demo, Expedition 33

**Distribution name and version where applicable**

GNOME OS Nightly

**Flatpak info**

flatpak --version → Flatpak 1.16.1
flatpak --gl-drivers → default, host

**Problem description**

When enabling Steam Input, the Steam client correctly recognizes the Xbox controller, but any games I run don't.

When Steam Input is disabled, games detect the controller and most of them use it correctly. But at least one (Hollow Knight) detects the wrong button distribution and doesn't even recognize some button presses. Curiously, others (e.g. Fatal Fury City of the Wolves) seem to work fine.

**Does this issue reproduce with native Steam**

Not tested, because native Steam is not installable in my demo.