While KLA is fundamentally flawed (DMA and even CV based cheats are becoming more popular as a result of KLA and they still give cheaters a significant (but now even harder to detect) advantage) it solves the problem of obvious and even most kinds of subtle cheating.
Attempting to detect cheating once inputs are being sent to your server (which is within your domain of control and on which you can implement non-intrusive anti-cheat) is very difficult to do reliably. An inexperienced player will make slow, delayed inputs. A highly experienced player will have reactions which are an order of magnitude faster (and in many cases faster than the speed of thought because of muscle memory). If you want to make a working but no longer detectable cheat, all you need to do is spend a bit of time and effort programming in human limits of reaction time into all your code and making sure all inputs look realistic (again, more limits).
At the end of the day, you can make a cheat which gives you a significant advantage without it being actually detectable by any statistical methods on the server.
At this point you might attempt to reach for AI but undoubtedly that will require human oversight or you will get false positives.
So, in summary, even if you were to design your game around server-side rendering and server-side input processing, forcing your players to effectively play over a remote desktop connection (which is impractical for any fast paced competitive multiplayer game due to latency issues but let's pretend those don't exist for a minute), you will still get cheaters with snap-to-head or recoil compensation or auto-fire making a significant impact on games. Heck, there's even the idea of using sounds (which need to be pretty accurate so human players can utilise them to determine where enemy players are) to implement a rudimentary wallhack.
This is just the nature of FPS games and why games companies end up implementing KLA for these games. The way to make an FPS game un-cheatable is to make a different game where cheating is more difficult or impossible just by nature of the format.
Want FPS without cheaters? Encourage people to do DIY matchmaking again, DIY server hosting and DIY administration. Except that "this doesn't scale". Neither does human review. Neither does server side rendering. The core reasons why game companies do KLA is that players will pay for games with KLA but won't pay for games without it. As much as I think Microsoft is one of the worst companies in existence, in this case I don't think they or KLA developers are to blame. KLA developers are simply doing what players want them to do and Microsoft is only allowing what their end-users want them to allow. If Microsoft removes KLA, it will be by replacing it with userspace code with hardware attestation support, it won't be by killing the concept of intrusive anti-cheat. All Microsoft is doing is trying to re-design the tools to cover their own ass.
Fundamentally, KLA has pushed cheating further into DMA and CV territory. This means that more obvious and annoying forms of cheating, undetectable by KLA are probably going to soon become more common. At this point the options are to have these games be console-only with blessed hardware and hardware attestation. And even that has flaws (as described). Eventually it will just be impossible to play a game in a public server without cheating. Maybe this will force people away from these types of games, or towards private lobbies. I don't know what the future holds here.