Yes and no. Yes, proof of works prevents double spending or more generally establishes consensus on which transaction become accepted. But no, this could be achieved with many different consensus protocol and does in general not require any irrecoverable loss. Just issue a key pair to each human and make them send signed votes to establish consensus by simple majority.

The true reason something like proof of work is required is because the system is also anonymous and distributed where being anonymous is the more critical aspect. Because of that you can not know who is out there and can cast a vote. Are this votes from a million legitimate users or did one user just send a million votes? The solution to prevent users from casting arbitrary many votes is then to make it hard for them to cast a vote, you make them for example waste a lot of clock cycles for each vote.

But, as I said, not because this is inherently a requirement but because you want to establish consensus among members of an anonymous group of unknown size and them wasting a certain number of clock cycles is a rough proxy for everyone gets one vote assuming that everyone has very roughly access to the same amount of computing power.