Update: A new Steam Beta Client should fix gamepad input in Big Picture Mode, thanks to Valve for the quick response.

The latest Steam Client update has left beta, bringing the Bluetooth Low Energy mode to everyone with the Steam Controller amongst other things. It also, sadly, broke other gamepad support in Big Picture mode.

There's this bug report on Valve's GitHub, which has been open since May 9th. That only mentions PS3 controllers, but there's other bug reports open for other gamepads. I've tested it myself, with the Steam Controller working as normal and all other gamepads are broken on Linux in Big Picture Mode.

This comes with a side effect, since this update is now stable, it's also broken everything but the SC in SteamOS as well.

It's not all doom and gloom and this update also pulls in some nice fixes, like these with the overlay:

Fixed an issue which caused the web browser to stop working in certain games

Fixed screenshots for games built with Unreal Engine 4 in OpenGL mode

Fixed some OpenGL API errors triggered by the Steam In-Game Overlay

Fixed a crash in some Vulkan apps when hiding the overlay or resizing windows

It also adds in Switch Pro Controller support, although I don't have one to test, I imagine that doesn't work for us due to the mentioned issues.

As for Linux-specific bugs, they fixed these:

Fixed a bug where trying to re-add old Steam Library folders to the client would fail with an error about the filesystem not being mounted with execute permissions

Fixed a common cause of games getting stuck when bringing up the overlay

Fixed a problem where trying to resize the client window would sometimes snap back to the original size

Fixed incorrect/rainbow colors when streaming from the desktop of a machine running recent Mesa

Fixed low resolution in Big Picture mode due to incorrect video memory detection on Mesa

See the full changelog here.