I used to admin on a BF2 server (many years back) and it was virtually impossible to catch someone in the act of cheating.
We could take remote screenshots to uncover any dodgy hacks etc, but I don't think I ever got someone. Problem was that some of the other admins were more gung-ho, and after I stood down I'm fairly sure they banned people for 'implausible K/D ratios'. Which IMO is ridiculous.
Yeah unless they are insta snapping to target or blatantly using a wallhack with no attempt to hide it getting solid proof is very difficult.
Its possible to be fairly sure for yourself if someone is cheating or not with enough experience and enough time watching them though - eventually they do screw up enough times you can be sure they are using a wallhack instead of just coincidence, or having a better understanding of the game mechanics than you do, etc. even then you have to be very aware of confirmation bias.
Another give away in FPS games is the connection between their aiming and strafe movement - most players will fine tune their aiming as much with the mouse as with their player position whereas if they are using software to assist aiming a large part of that doesn't happen - however you will sometimes encounter a high level player who has truly separated their movement and aiming cognisance.
When it comes to cheating stats mean pretty much nothing - you will often find people with truly tragic stats using a wallhack, etc. and while they generally suck so badly anyhow the impact of that on other players is generally minimal it is still cheating and will still sometimes get them kills that they shouldn't.
Sadly there is very little that can be done other than having well admined dedicated servers as the cheat authors are almost always one step ahead of the game developers and can dedicate more time to it as well.