Phase locked Loop in Demodulation
The key here is this statement "My understanding is that a PLL is used to demodulate in situations when the demodulator knows the carrier frequency but does not know the phase."

There is only one small insight that you are lacking:

Lets say we have an input waveform \$ \sin ( \omega t - \phi_1) \$ and the output from the VCO which is frequency locked but out of phase \$ \sin ( \omega t - \phi_2) \$ but in the second waveform I have a way of changing the phase (we'll ignore HOW for now) so the second waveform becomes \$ \sin ( \omega t - \phi_2 +At) \$ clearing up the second waveform expression gives us \$ \sin ( (\omega+A)t - \phi_2) \$.  From this you can see that an instantaneous change in phase is actually a change in frequency.  Conversely you can also express a the difference in frequencies of two waveforms as being two waveforms that are at the same frequency but one has a time varying phase.

Frequency and phase are really just two sides of the same coin.  Clearly if the frequencies are far apart it doesn't make sense to talk about differing phase.  Also once the frequencies are close together or even locked then it does not make sense to talk about differing frequency.

However, the modulator/LPF combination is a phase detector that behaves well (i.e. gives the right signal direction - error voltage) and allows the VCO to slew in frequency until it gets close.  In short it can't know the frequency w/o knowing the phase.

A good phase detector will have a sigmoid shaped response curve.  It will saturate high when the VCO frequency is way to low, it will saturate low when the VCO frequency is way too high and at some point when it is close in frequency and it makes sense to be talking about phase, then it should have a nice linear curve that is an odd function.  You could view it that the curve is what changes the Modulator/LPF combo from a frequency to a phase detector.