EAC anticheat does work on Linux, but it requires a little effort from the developer to enable it, for example I played New World on Linux which requires EAC anticheat.
As for Blizzard I have no idea about the other Blizzard games, but I play Diablo 4 using Lutris, other than one update that caused the Battle.net launcher problems it works fine, at least for D4 (and D3).