How do we know certain aspects of QM are unknowable?

It isn't that the position of the electron is unknown, but rather that the electron simply doesn't have a position.

The idea that an object has a position is such an intuitive one that it's hard to believe this might not be true. But a position, in the sense of a precisely defined location where we can find an object, is a macroscopic concept that simply doesn't exist for quantum objects. An electron in a hydrogen atom is delocalised and does not have a position for us to know.

In the early days of QM it was suggested that parameters like position do really exist but that we can't know them. This was known as the hidden variables theory. However as Robin suggests in a comment a theorem due to John Stewart Bell can be used to test whether hidden variables are consistent with observation, and to date the results suggest that the hidden variable theory is not consistent with experiment.

To address your last paragraph: quantum mechanics is a mathematical model that seems to describe the real world very well. Extraordinarily well in fact. If you're asking whether probability distributions really describe particles then you'll have to ask a philosopher what the word really means. All we know is that the predictions we make using quantum mechanics match experiment.