I could be wrong but I believe most games (e.g: Fairfight in all EA games) do have a server-side anti-cheats, just not very effective ones because they like for EAC/BE to do most of the work.
using LUA is an interesting idea to say the least but again likely wouldn't prevent or demotivate current cheaters from doing their thing.
definitely not all cheaters are sociopaths, been in the cheating community since 2014-ish (starting to slowly back off and leave recently just due to the the obvious toxicity in it) and have made plenty of genuinely great friends there but ofc it is mostly horrible people, especially the people who closet and hide it from friends. do agree that this plan would never work though, but HvH was never really dead, it was at its peak in 2020 where there were probably 200+ people playing although it's really only active in EU now.
this might (barely) work for Source games like CS:GO, TF2, etc but on games like Overwatch, Valorant, Fortnite, etc there are a lot less exploits that would keep HvH actually interestering like fake angles, doubletap, etc like there are in the Source Engine (mind you this is because the engine is literally from 2003 and based on Goldsrc which is based on the damn Quake engine.)
Correction, EAC barely works on linux. Apex is just safer because Respawn themselves are putting in some effort.