Anticheat isnt solely about kernel calls. Anticheat systems, depending on what one you are referring to, will inspect runtime memory, data loaded into RAM. It will do a number of things to verify memory isn’t being modified (which cheat engines, among other things, need to do).

Simply, Wine and linux load applications differently, anticheat systems see the difference and assume something nefarious is going on.

Its not as simple as just running anticheat in wine.

edit some additional info from a pretty old article

https://www.theverge.com/2021/10/5/22709918/valve-steam-deck-supported-games-anti-cheat-proton-eac-battleye-epic

Also, (and this is from security research articles here) most kernel level anticheats seem to focus more on datamining than anticheat (see: anything from tencent)

Its so bad that a lot of corporate environments ban any work being done on machines that also have them installed (source: my employer)

Over time more and more anricheat companies have realised that personal data is gold and they are harvesting more and more of it.

Just read the eula some time. Most of it state in plain english that they send files from your documents, take screenshots and log keys.

And we give them kernel access…

Because it is in user space not kernel space. Also Linux uses a monolithic kernel (no separate space for drivers)