> why incorporate it instead of just having proper moderation and/or private servers?

No one wants to become a moderator, they do it out of necessity. So it's pretty much the other way around: a lot of anticheats were, and are, originally developed by community members for private servers (because you're not deploying a 3rd party anti-cheat onto first party servers). BattleEye was originally for Battlefield games. Punkbuster for Team Fortress. EasyAntiCheat for Counter Strike. I even remember Starcraft Brood War 3rd party server ICCUP with a custom 'anti-hack' client requirement.

You still see this today with Counter Strike 2 private servers Face-IT: they have additional anti-cheat not less. Same with GTA V modded private server, FiveM have anti-cheat they call adhesive.

And then game developer saw that players are doing that, so they integrate the anti-cheat so that players do not have to go downloading/installing the anti-cheat separately. Quake 3 Arena added Punkbuster in an update for example.