Most is a stretch.

The FPS in the top 100 currently most popular games on Steam[0] and their status [1]:

  #01 CS:GO - native
  #04 PUBG - anticheat
  #05 CoD MW2 - anticheat
  #06 Apex - Works (it has anticheat that works on Linux)
  #07 TF2 - native
  #09 Rust - Works
  #12 Destiny 2 - anticheat
  #21 Rainbow 6 - anticheat
  #22 DayZ - Works (it has anticheat that works on Linux)
  #26 Warframe - Works
  #69 Payday2 - Works
  #78 Arma 3 - Works (it has anticheat that works on Linux)
  #79 CS:S - Native

Native or working: 9/13

Broken: 4/13

Non-steam or outside top-100:

  OW2 - Linux is second class and not actively supported - but Blizzard have unblocked Linux support when issues were reported
  Battlefield (all?) - Works
  CoD (before MW2) - Works
  Gundam Evolution - anticheat

[0] https://store.steampowered.com/charts/mostplayed

[1] https://www.protondb.com/

I'm probably spectacularly unlucky as I play Siege, Destiny 2, CoD MW2, Hunt Showdown (#63), and the occasional Fortnite (EGS) and Valorant (Riot), none of which work.

Battlefield 2042 is on Steam but not supported, though I'm not sure if that's the anti-cheat solution or Proton.

Sadly even a few top games not running on Linux is enough to keep most PC gamers off of Linux.

Since the announcement of the Steam Deck, Valve has made various promises to work with the AC providers to bring support to Linux. So far they have brought support for Epic's EAC, and it seems they are working on bringing it for other titles as well.

An increasing number of games with Anti-cheat work.

https://areweanticheatyet.com