Update 08/11/23, 18:31 UTC - They just put out a patch that solves it. Over a week to put back a single removed file. I groan, but at least it's sorted now.
One of the most popular titles on Steam, Apex Legends, has been left broken on Linux / Steam Deck for over a week now after the Ignite Update. You can fix it yourself though, if you want to risk it.
The problem: they seem to have accidentally removed the Easy Anti-Cheat .so file that it needs to work with Proton, so it just doesn't let you in due to it not detecting that the anti-cheat is actually there (because it isn't). Somehow over a week later EA / Respawn haven't just put it back.
Respawn have confirmed they're looking into it, as-per their Trello page that currently lists the fix as "In Progress", but it's been under the same status since they added it to their list on October 31st. For a game that's Steam Deck Verified, this is quite a nuisance. And since it's not the first time it's happened - you have to wonder why it's taking so long.
If you want to fix it yourself at your own risk, you just need to grab a copy of the missing file (like this one on Reddit from when it happened before) and place it back into the installed folder. The location depends on what Steam Library you have it installed in somewhere like
This will allow it to work again but I don't recommend it, always best to wait for their official fix. But if you're desperate to play at least it works.