A lot of (not linux-savvy) people claim SteamOS will be the solution to all.
I have hope, but not because SteamOS will be some revolutionary thing, but that maybe we'll get a proper chain of trust mechanism that can make real anti-cheat possible.
Steam is a good authority that could push kernel and driver signing forward, which would be a huge deal for making effective AC solutions that possibly don't even need kernel access. IMHO, popular distro distributors / packagers would be a good enough source of trust.
For example, Arch's linux packages could be signed by Arch Linux. This signature would be accepted by ACs, and could be "un-accepted" at any point if the developer would spot Arch doing something shady (unlikely). I see no reason why Microsoft should be the only "TRUSTWORTHY" authority. Do you really think Arch would take a bribe to add "my epic le fortnite hax" to their kernel build?!
With a properly signed kernel, and no unsigned kernel modules allowed, we should be able to get a pretty trustworthy kernel space. In contrast to windows, most drivers in Linux are in the kernel, so it would take only one signature to sign them all.
Without this being standard, the kernel space is just too easy to compromise, at which point developing working anti-cheat is extremely hard (on the client side). There are FOSS kernel modules available on Github that can neuter userspace anticheats.
It's already been shown that companies are unwilling to implement proper server-side AC. Even though it's much more privacy-respecting, it's less effective overall, and harder to develop other than simple rage-cheat detection. There's simply no way to detect a very well-written cheat that gives you a slight advantage on the server side, while in many games that slight advantage could translate to a much higher win ratio.
The key, in my opinion, to a working client-side AC is a kernel that cannot be modified, even by the AC itself. By nothing, bar a bug in the implementation that can be fixed. Nothing gets into the kernel other than the signed kernel itself. We then get a nice, userspace arms race between AC developers and cheat developers. The arms race is unavoidable, but the nice part is that it's in userspace for once.