Negative feedback and op-amp poles
But I was talking to someone who insists that the pole is still down at 300Hz causing significant phase shift before we even get into the kHz.

Of course physically the circuit still has a pole at 300 Hz. However, through the use of feedback we can make the pole of the circuit (as we see it from the outside) move to a different frequency at the cost of gain.

About the phase shift, what this person forgets is that phase and frequency are related. A 90 degrees phase shift at 1 MHz is 250 ns, while at 300 Hz that 250 ns is only 0.027 degrees.

Also the phase shift is compensated for by the feedback loop. So:
...significant phase shift before we even get into the kHz is true but that is irrelevant as the loop is compensating for that phase shift.

If the phase shift from the 300 Hz pole was an issue then it would be impossible to apply the feedback and still get a stable amplifier.

You might want to ask this person who insists about this phase shift to explain why amplifiers with feedback can still be stable. If there is so much phase shift at a few kHz then how can the system be stable? (it cannot, that person doesn't understand how feedback works)