It is a trivial fact that on a classical computer, you can simulate a quantum computer in time that grows linearly in circuit depth, in principle (as in the case of the "naive" way I mentioned above). No one in doing quantum algrothims ever claimed this was the case, and claiming otherwise is just a silly mistake.

Preskill coined the word "quantum supremacy", not Martinis. Even if someone from Martinis' lab misspoke, you can't pin it on Preskill.

Take Shor's algorithm, which has been the poster boy of quantum computing for decades. It gives exponential speed up in the number of qubits, not circuit depth.

See other examples here: https://quantumalgorithmzoo.org/ The complexity is in the number of qubits, "cirucit depth" is a non-concept in quantum algorithms.

No one cares about the circuit depth in quantum algorithms, because in principle, you can always reduce the circuit depth to 1 on a quantum computer: quantum computers aren't necessarily made of basic logic gates, you can implement any unitary in by for example smoothly pulsing the fields in a correct way in a single go.

"Depth" is a concept which usually comes into play when you try to measure the quality of the implementation of some given gate U. You repeat U many many times say M, and see how some measured quantity decays as a function of M, which gives a measure of the fidelity. But this has nothing to do with the quantum algorithm's complexity, it's just for "benchmarking" a physical implementation of the gate.