Doubt about Phase Locked Loop
Simplistically, the loop filter can contain an integrating element, so that the output of the phase detector going to zero doesn't necessarily mean the control voltage of the VCO will be zero.

To make things more complex, remember that phase is the integral of frequency, so by using a phase detector rather than a frequency detector, you've already included an integrating device in your control loop. The loop filter will likely need to include both integrating and differentiating components to keep the loop stable, while also allowing the loop to react quickly to disturbances in the VCO behavior, thus minimizing phase noise of the output.