Inconsistent Controller Detection

**Game information**
Dragon Age 2 (Proton)
Dark Messiah of Might and Magic (Proton)
Overlord II (native)

**Distribution name and version where applicable**
Fedora Silverblue 33

**Flatpak info**
Flatpak 1.8.2
default
host

**Problem description**
Dualshock 4 is not being detected in many games as of recently. It used to work flawlessly. The Steam Big Picture UI detects the controller properly and navigation through the system works wonderfully well. However, as of yesterday, the controller has not been detected in Dragon Age 2, Overlord II, and Dark Messiah of Might and Magic. RetroArch (proton) detects the controller as an xbox 360 controller and it works with it just fine.

**Does this issue reproduce with native Steam**
Have not tested

The report names Dragon Age 2, Dark Messiah of Might and Magic, Overlord II, Steam Big Picture, RetroArch, and the Fedora Silverblue/Flatpak environment, but no repository file or test. Reproduce the controller-detection difference across those games and compare it with native Steam; done means identifying and fixing the regression so the DualShock 4 is detected consistently.

Written by the indexing model from the issue text.