Not a true integrator, just a lowpass filter. The VCO is the true integrator.
Hey, I just said that!
The vco turns voltage into frequency, but the phase detector detects, well phase. Phase is the integral of frequency. It's the combination of VCO+Phase detector that becomes a voltage-to-voltage integral.
Most VCOs, specifically VCXOs, have a lowpass filter between the VCO analog input and the varicap. They really have to. That is typically in the 10 KHz range. So that lowpass filter cascaded with the inherent VCO phase integration makes that block 2nd order. The lowpass filter after the D-flop makes the loop 3rd order. But if KVCO is small (as it is for the usual VCXO with a small pull range) it's easy to have the integral dominate, and get a 1st order, dead stable loop. Of course, things like this have narrow acquire and lock ranges. Wideband PLLs are a lot different.
One thing that we sometimes do is to build a triggerable start/stop oscillator and, after it's triggered, lock it to a crystal oscillator reference, but keep the start/stop oscillator edges time-coherent to the trigger. The math can get boggling.