There already are games with anti-cheat on Linux, they're just using (arguably) less effective versions of them.
Valve are financially incentivized to get as many games working as possible, I don't think it makes sense for them to do nothing at all about this. They have a head start now due to the general crappiness of Windows on a handheld but Microsoft's mobile offerings could catch up and given that would have Call of Duty among a lot of other heavy hitters available while Deck wouldn't, that is a problem that Valve would need to solve in order to not get stomped.
Valve are financially incentivized to get as many games working as possible, I don't think it makes sense for them to do nothing at all about this. They have a head start now due to the general crappiness of Windows on a handheld but Microsoft's mobile offerings could catch up and given that would have Call of Duty among a lot of other heavy hitters available while Deck wouldn't, that is a problem that Valve would need to solve in order to not get stomped.