As far as I know, no one has ever gotten their account banned store-wide.
Like you mentioned, sometimes games with anti-cheats like Apex Legends or Overwatch will falsely ban linux users (from that specific game - not the whole Battle.net or EA store), but they usually undo those bans within a week or so for most users.
I was hit with one of the Apex bans, and it was overturned within a week or two. But there were a few people who I am pretty convinced did not cheat and are still banned, so maybe be cautious with that particular game.