It's not illegal at all. Sports league are private entities and to compete in them, clubs have to abide by their rules. What you're saying simply doesn't apply.
In other sports, for instance, the NFL, there is a salary cap. No team is allowed to just spend their money as much as they want. It's not illegal, it's part of the rules that clubs must follow to gain the benefits of competing in the league.
There is zero chance City can sue and have the PL repay them. That take is simply not understanding how entities like sports leagues operate.