Is a worker's labor power the private property of the worker?

Hobbes was a fag, natural rights are a meme.

Nobody is talking about natural rights, you ignoramus. They're talking about private property, which is a material relationship between people, disguised as a mutual relationship toward objects.

Rent stabilizes at the opportunity cost of not engaging in a transaction. Surplus extraction is capitalists renting MoP out to workers.

B-but muh self ownership!

False. In no circumstance does "rent" entail automatic ownership of anything produced with the rented property. Both rent and surplus value are unearned income; that's where their similarities begin and end.

Rent is much more general than you think it is. Surplus value extraction is a subset of it.

If the employer-employee relationship were rent, it would be possible for workers to make a net loss by working (rental rate exceeding revenues within a certain time period). This is, however, impossible.
Firms rent out capital to other firms all the time, and it's always a rate for time rented, never a situation in which the rental firm takes ownership of the entire proceeds of production less some fixed payment.

Then the workers would just quit. They pay the rent in labor value.

The premise is false; workers do not rent the means of production. In fact, private property is the inverse of rental.
When someone rents property from another, they can keep whatever they make with it, but must make a fixed payment to use the property.
When someone uses another's private property, they cannot keep whatever they make with it (the owner does), but they receive a fixed payment (wages).