I did some machine repairs long ago. In VCRs/VTRs the control track is especially critical, since the video head alignment is dependent of it.
Some analog alternatives are to design an analog phase-locked-loop (which is not trivial, and I believe what a typical VTR uses) or a more amicable pwm to DC circuit. A sample and hold of an integrating ramp would be very precise. The sample and hold uses the perforating pulse to reset an integrating circuit (a ramp) and the hold keeps the ramp voltage before is reset.
One edge of the pulse can reset a 16 bit counter, and it will provide a precise period. For the output, a DAC (probably 16 bit too) can then be used to provide the control output to the servo. When the period is lower/higher than the set period, the output would increase/decrease accordingly. This should be doable as long as the counter and register have enough resolution, my guess is that 16 bit would be a must.
If the 16 bit DAC is not handy, a 16 bit PWM output can be an alternative, with a low-pass RC to convert to DC. For a clean DC output, the response time would be a bit slow, but it should work too. I used it to control the constant-current driver, and this posting has relevant references for the PWM workaround to get 16 bit out of an Uno.