I also just realized that qiskit.providers.basicaer.UnitarySimulatorPy.run() (https://github.com/Qiskit/qiskit/blob/main/qiskit/providers/basicaer/unitary_simulator.py#L206) never deprecated a Qobj input and will still support it while the qasm simulator and statevector simulator will not. I still think we should move forward with your PR, but it's something we'll want to clean up in a subsequent PR.

Originally posted by @mtreinish in #10872 (review)