I think phase margin is more intuitive for the stability. Cause we can easily know that the loop gain at the cross over frequency(f1) is:
T(j*f1)=-exp(j P) where P is the phase margin here.
the error function at f1 then be:
1/(1+1/T(j*f1))=1/(1--exp(j P))
We can easily calculated that only P<60 will induce a error greater than one at f1, that is the peaking begining. In fact, the peaking occurs for P<65 and ringing happen when P<75.
However, from Gain margin, it is not that easily to know the peaking and ringing that intuitive.
So I think that is why they always use the Phase marging here.