With Splitgate moving over to player hosted matches with the main servers being shut, the developers have now removed the Native Linux version too.
Going by SteamDB, it appears this happened a few days ago. So now if you wish to continue playing the classic Splitgate, you'll have to use the Windows version with Proton. Since Valve's verification for Steam Deck was done on the Native Linux version that no longer exists, you will need to manually change it over to Proton in the compatibility settings otherwise it won't launch anymore on Linux / SteamOS and Steam Deck.
Looks like the anti-cheat was removed as well.
It's a shame whenever this happens, but in this case it's a free to play online game that is no longer supported. So, running it with Proton is likely the better option compared with an old Linux version. At least Valve can maintain compatibility with Proton in the case were future Linux updates might just leave the Native version broken.