I think that quantum physicists do get an (imperfect) intuition for it, but they can't communicate it to laypeople because the inferential distance is too high.

I also think that it takes awhile for societies to internalize these things, and to find what works pedagogically, so it's hard to say how intuitive it will end up being. For example, starting teaching with quantum computer science might be really beneficial because qubits bypass some of the difficulties (differential equations, waves vs particles, tunneling) while keeping most of the weird (superposition, entanglement, measurement, interference, counterfactuals).