I completely agree with you that from the perspective of a quantum information theorist computing the spectrum of the hydrogen atom is a rather complicated thing. I disagree wholeheartedly that this is part of the essence of quantum mechanics.

The hydrogen atom is one system, understanding conceptually that its behavior is governed by a self-adjoint operator and its spectrum is very relevant to the whole of quantum physics. Understanding exactly the details of the calculation I think are not. Especially because if you do the calculation within quantum mechanics without quantum field theory you will obtain a somewhat incorrect result anyway (you will miss interesting phenomena like Lamb shift).

Similarly interference is an interesting phenomenon that one needs to understand to understand quantum mechanics, but understanding the specific calculation of how interference makes nice patterns in some example setup isn't particularly enlightening.

I agree that the Heisenberg uncertainty principle is important, but it certainly be derived from Aaronson's point of view (e.g. the standard Robertson-Schrödinger inequality is easily obtained).

As an aside, I also think that self-adjoint operators and the correspondence principle are a fairly terrible way to think about observables in quantum mechanics. An obvious fact is that every measurement of (for example) the position of a particle has some unavoidable experimental error (our apparatus only has finite resolution) so the thing we actually measure in reality is some fuzzy observable which can not be represented as a self-adjoint operator. A POVM is a much more natural candidate (as a physicist it is natural to assume that the thing you get by adding some classical noise to your measurement is still a measurement).