The upcoming skate. from EA will be unsupported on Linux / SteamOS and Steam Deck due to anti-cheat

No, but I think they do it anyway as a preemptive tactic to reduce the share of users on Linux as much as possible.

Every time there is a small wave of users that transition to Linux it means less market share for Microsoft. The more users that get on Linux and have a pleasant experience and things just work means a better reputation and word of mouth spreading between people which translates into a greater potential for even more people to make the switch.

Things like this start slow but once they start to gain traction it can turn into a much bigger problem for Microsoft and they know that. So it would essentially be them trying to "nip it in the bud" before it becomes something that they are too far behind on to get control of.

Eh, you could make your own kernel module and require it being enabled for Linux as well. It would be a major undertaking though, and since the linux share is very small still it just isn't worth it.