The characteristic trait of quantum mechanics

Erwin Schroedinger introduced the term "entanglement" with this 1935 paper:

1. When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or ψ-functions) have become entangled. To disentangle them we must gather further information by experiment, although we knew as much as anybody could possibly know about all that happened.

This is an important insight, but I don't agree that this is really "the characteristic trait of quantum mechanics.

How is it that quantum mechanics allows creating systems where we could know "as much as anybody could possibly know", and still leave some questions unanswered?

In order for entanglement to seem so mysterious, it has to be combined with some other quantum mystery.

I have argued here that the the characteristic trait of quantum mechanics is the non-commuting observables.

Sure enough, Schroedinger’s argument in the next few pages depends on non-commuting observables. That is where the quantum weirdness is. It is not so weird that our knowledge of a system could depend on a system with which it previously interacted.