Yes, you can see phase of oscillation frequency ( and its' harmonics) by PSS (or HB simulation)
But this phase will be steady state phase, in fact the phase will be stalled at the beginning of the oscillation and the is settled downby time.
Unfortunately, you can not see initial phase swingings..
Re: simulation method of the oscillated frequency output pha
Maybe a quick and dirty method: simulate two similar oscillators, let the first one start up at t=0 and the second one after a certain delay. Multiply the output waveforms using mathematical IQ demodulation and extract the phase. Because it is a simulation the oscillation frequency of both oscillators will be exactly the same, so it should work. Then you'll see the startup phase transient of the oscillator.
It's likely that you'll see a frequency shift first (= very fast phase transient) and then a slower settling phase transient.