Genuinely curious, because this isn't my area of expertise, but how do you design a server to be "better" if it has to trust data from a remote client?

Check the data on the server ("oh no, incredibly expensive"). Don't give any data to the client it doesn't need, like enemies around the corner ("oh no, now my game is so very laggy because caching and future position assumption just became impossible")

Example, if the client is compromised - because as they've said, they have no way to "attest" that the kernel is not compromised - how would the server know any better?

Now the server doesn't need to care. There's input? Validate and use it.

If my Apex client tells the server I got a perfect headshot, how would the server know I didn't fake the data? Is there a real answer to this problem or are we just wishing they come up with an impossible solution?

Now the client can go pound sand. Server decides if it's a headshot. Client only sends coordinates of origin and target. Lag? Sucks to be you, with or without cheat.

My general understanding is that EA is 100% correct. Now, on the other hand, maybe the should just limit plays between Linux <-> Linux so people can at least still enjoy the game

That would only create more work for the developers, all for the defacto expulsion of Linux users (Way less players at all times). The best course of action here would be the actual expulsion of Linux users. Also, EA is at most 25% correct. (Not a rational argument, I just very much dislike them)

(I'm moving to Linux soon so I'll basically no longer be able to play the game, which is, as my primary gaming addiction, a huge loss I'm willing to take).

Damn, sorry to hear that. It's always bad to leave something one knows because something's become unbearable. I wish you best of luck on your journey! (I'm assuming a lot, but why else would you switch despite your choice of use of free time?)

There's compromises EA could take, but I think the Linux market share is just too small for them to care to spend any resources - even though they're raking in billions (~$3.4 Billion) and could spare a few resources to find a good middle ground. Capitalism at it's finest.

On the other hand: I quite like it. It forces them to keep their grubby little hands from my kernel.

I do not like anything anti cheat. But I also don't really like cheaters, especially in online games, so anti cheat could be tolerated. The only thing is: nothing trumps my systems integrity. Definitely not online player satisfaction.