I think it's perfectly fair for the prof to catch students who are cheating this way.
i don't think it's very clear cut or obviously perfectly fair.
I don't think this matters. Cheating was against the rules and a cheater demonstrates a failure to adhere to a social contract and voluntarily waives their right to be assessed the same with those who did follow the rules. At least there are avenues for a cheater to have another attempt, even if they have to move to another institution.