That's true. But if you can't prove that your hash function has the desirable properties you expect/hope for, then I think that is important to acknowledge.
No, those are separate issues. There might be philosophical value on acknowledging the impossibility to prove the correctness of the algorithm, but it has little or no practical value when you want to prove if your implementation of the algorithm is correct.
Your argument is like saying that because there is no way to guarantee your physical product's design is 100% defect free, there is no point to implement quality controls in the manufacturing floor.
Your argument is like saying that because there is no way to guarantee your physical product's design is 100% defect free, there is no point to implement quality controls in the manufacturing floor.