The crossover frequency impacts the fastness? What about the 3d point?
I feel that some terms are used too vaguely in this thread.
Phase and gain margin and crossover frequency are parameters of the
loop gain of a feedback system. It's only identical with the
open loop gain of an amplifier for a feedback factor of -1, on in other words a
closed loop gain of +1. Unfortunately feedback factor wasn't mentioned explicitly in this thread, it's only implied in post #2.
"3 dB point" can be read as synonym for the closed loop bandwidth of a feedback system with intentionally flat gain characteristic respectively frequency independent feedback factor, e.g. said -1. It's related to the loop unity gain bandwidth (or cross over frequency), but not the same. For a system with sufficient phase margin, both frequencies are close together. The smaller the phase margin, the higher the gain peaking at the crossover frequency, resulting in an increase of closed loop bandwidth.
We check the closed loop stability by looking at the loop gain's phase and gain margins, in other words, if the loop gain has any positive phase margin (even by 1º) and gain margin > 1, the closed loop is stable.
Phase margin impacts the damping of the system and the crossover frequency impacts the fastness of the system (closed loop, obviously).
The loop gain can have multiple crossover frequencies, in this case the simplified (Barkhausen) stability criterion isn't applicable. You have to evaluate the full Nyquist stability criterion.
Suggestion, review previous Edaboard threads about feedback system calculation, particularly contributions of senior expert LvW.