Loop Gain and Phase Margin Correlation
Now, Aβ would typically be a value ranging 1000000 to 10000 (in opamp
amplifier systems, where open-loop gain is usually around 120dB).

That's at DC where nobody really worries about phase margin because it's never going to be an issue. Look at a typical open-loop response of an op-amp: -

Picture source and other relevant information that could be useful.
At 1 kHz the open loop response might have dropped to 60 dB (G = 1000). At 1  MHz, the o/l gain is only 10 and this is an area where quite a few op-amp circuits have problems. Arctan of 10 (assuming a unity gain situation) is 84 degrees and consistent with the graph above. At 10 MHz the gain is unity and arctan of 1 is 45 degrees i.e. not a million miles off.
If you know the T.F. of the forward gain device and you know the feedback T.F. then certainly you can calculate phase margin by considering the loop broken with a signal being injected at the input and the output being taken from where the break is. But you have to respect impedances and loading when you open the loop and sometimes it can be difficult to realize.