Your system information
Steam client version: May 12 2022, at 20:36:43
Distribution: Manjaro KDE Plasma 21.2.6
Opted into Steam client beta?: No
Have you checked for system updates?: Yes
Just had the exact same issue in Manjaro. This was working fine a week ago. But not after a complete system reinstall. I set up non-Steam shortcuts for both Firefox and RetroArch (run as a flatpak) and both failed to launch with the updating dialog. In offline mode they failed to launch entirely.
Turning off "Enable Steam Play for all other Titles" fixed the issue.
It's almost as if Proton isn't distinguishing between a Linux Native application and Windows exe, possibly trying to run the Linux App as a Windows exe.
Your system information
Steam client version: 1654574690
Distribution: Endeavour OS (Arch based)
Opted into Steam client beta?: No
Have you checked for system updates?: Yes
Issue is still present in the latest version of the client.
Can reproduce on Endeavour OS.
Have the same issue with the latest Steam client on Arch.
Steam client version: 1660960119
Distribution: Arch Linux
Opted into Steam client beta?: No
Have you checked for system updates?: Yes
Nothing extracted yet.
Your system information
Please describe your issue in as much detail as possible:
When launching a non-Steam Game with "Enable Steam Play for all other Titles", Steam tries to Update the non-Steam Game

Using the Fedora RPM Version, this never completes, while it succeeds in the .deb PopOS Version
Steps for reproducing this issue: