Phase and frequency are the same information, one is the derivative of the other.
Often in a PLL, the output is fed through a frequency divider or is heterodyned before being applied to a phase comparator, which is operating at a different frequency from the output. That phase is definitely not the integral of the output frequency and is not "the same information".
I could also apply the PLL output to a simple frequency to voltage converter and apply that voltage to a standard op amp error amplifier. Its output could drive a V to F converter to generate the PLL output.
I could go on. But the point is that controlling signal type A in no way requires you to feed back signal type A to the error amplifier.
The feedback path distinction being debated here is essentially about impedance; when it's high it's being called VFB and when low it's being called CFB. Where is the boundary?
I disagree. Impedance magnitude is associated with certain CFA and VFA input stages, but it is hardly the main distinguishing feature. Among the distinguishing features are bandwidth variation over a certain range of closed loop gains, and slew rate assist on demand.
I have built out of discrete components two otherwise very similar amplifiers, one a CFA, and the other a VFA. They clearly demonstrate the bandwidth variations I mentioned. Please see part 1 of https://www.audioxpress.com/article/current-feedback-fake-news-or-the-real-deal .