For the same reason public-key signatures are worth trusting: if correctly designed and validated, it's computationally intractable to construct one without the

For the same reason public-key signatures are worth trusting: if correctly designed and validated, it's computationally intractable to construct one without the information (private key or otherwise) being proved. (You do lose the inability of coparties to prove that you know the thing, but that's not always a problem.)