DataDrake Blizzard isn't banning Linux users.
There have been a few ban waves affecting Linux users in the past, but the bans of legitimate users (there were some Linux users who were cheating / using macro scripts) were reverted. Personally, I've never been banned.
I game [nearly] exclusively on Solus (I only use W10 to play Halo MCC multiplayer). Overwatch works, I've been playing it on and off (I don't play it too often nowadays) on Linux for over 2 years without issue.
WoW / WoW Classic works fine as well, I only stopped playing it because I don't want to pay a subscription to a company that's supporting China's regime.