Sending away all of your PII to be verified in games isn’t much better (in terms of intrusiveness) than running anti-cheat software in the first place.

Additionally, will it be any harder to get around a system like this than a traditional anti-cheat system? Theoretically - for the average player at least, it might make cheating worse - assuming the cheater runs their cheats in a realistic way, in your world, there is now no detection software pinpointing the cheaters!