• In Steam, press the cogwheel (settings) and go to "Properties".
  
• Under the "Compatibility" tab, select "Proton 8.0-5" to use this version of Proton.
  
• Relaunch the game and it shouldn't crash anymore

I tried experimental and 9.0-1, but neither of those worked for me, so just pick 8.0-5.

So what i THINK happens here, is you've played the game before a few years back through EA launcher/origin and your cloud save is still bound to your EA account.
EA took down the servers, but they didn't take any countermeasures to ensure the game would work, so the game will infinitely try to connect to the disabled servers to sync up your cloud save.
(I think this is the case because when i launched the game for the first time on steam/my deck, it gave me TONS of achievements from my old EA playthrough).

• Logging in to another EA account isn't an option, because it'll tell you the game isn't activated on that account.
  
• Turning off your internet before launching won't work, because the EA app will refuse to boot the game unless you log in
  (which also means you can't play without internet, wohoo).

Here's how i bypassed it;

• I fully uninstalled the game from steam and made sure there were no folders left under steamapps or documents (there weren't).
  
• I re-downloaded the game through steam
  
• On my first time launch, i logged in to my main EA account (containing Mirror's edge and the broken cloud save).
  
• I let EA log in, do it's thing and boot the game
  
• As soon as the game booted (white screen followed by brightness select slider), i turned off wifi through the steam quick menu (3 dots button under the right trackpad)
  
• The game now started the cutscene like before, but this time i could hold A to skip it and put me in control of Faith, starting the actual gameplay.

I'm not sure if a full uninstall of the game was neccesary, i just assumed the game would "remember" that it has a cloud save to load every time, you could try the wifi toggle without a full uninstall first.

I hope this helps.
EA should really at the very least remove the need for the online servers, because this is ridiculous. If anything i hope they pin this post to help future Steam Deck users.