Can someone explain to me why cheating shouldn’t be solely caught and prevented on the servers these games run on? Why the hell are they diving into control on my own system?

It’s all about control. They demand control over servers, not allowing you to self host. They also demand control over communities, by putting them all in one big server or other controlled channels (but sometimes refusing to properly moderate them like in league of legends).

Now, they want to control your device.