How fast can a Quartz Clock spin? 

you need to reverse the current flow in the solenoid to get the stepper to step. If you apply a true, 50hz AC signal, this is automatic, even though cross over between +ve and -ve voltage can't be controlled.... however, doing the same thing with an h-bridge and you can certainly drive the solenoid beyond what the video suggests.

Btw. With the right controller logic - speed adapted - you can get pretty amazing speeds on those motors. I don't know where you live but in Europe we have the DCF transmitter and if you want a radio controlled movement fitting "euro hands" - typically for new German clock and vintage European clocks of many brands - if you want to preserve the original hands unaltered (you can even keep original movement to reverse is possible) your only option is the Hechinger W70 RC clock movement. It's a one motor movement (many RC today is two motors) with step each second. The amazing thing is when it adjusts itself - which it even in worse case granted good radio reception only takes 15 minutes. 3 minutes for reception - 12 for setting. The second hand literally races around at a speed of approx. one revolution pr. second. A have seen a few other one motor RC clocks before including the before mentioned but that is pretty slow using about 5-6 minutes to gain one hour - which the Hechinger does in only minute.

in the case of the atomic clock, how did you detect home position for the hands?

Smartphone David thanks for corroborating, always nice to remember that not all youtube comments are mean

It would have been great to see the hands spin. Maybe it would have affected the top speed too.

You may be right. I assumed that the limiting factor would be the physical resistance to spin of the pinion gear. However, it may simply have been that my transistors could not switch fast enough. Though I still do find that idea surprising: I would have thought that even these transistors should be able to switch hundreds or even thousands of times a second (I'll double check that now :) ). Looking at your experience though I suspect you may know something I don't. Can you please post more of your thinking here so that others can learn from it too?

I doubt the second hand would be a very good flywheel because it is spinning at about 1/30th the motor speed. I suspect highest speed would be obtained with no load and the permanent magnet rotor behaving as the flywheel. Modifying the second hand to increase its weight (angular inertia) without ruining it's balance may help its behaving as a flywheel.

I was able to get the second hand to do a rotation in 3 seconds reliably. I did with other code achieve the 6 second rotation I was looking for. Now I have to figure out how to get this gimmick clock to reset to 12:00 position, start on a button press, run for 6 real minutes ( minute hand will travel 1 full revolution) and stop. cycle will repeat as needed. The timing is so critical, I spent a long time trying small changes and still can't find the sweet spot for running 6x faster than normal with a Lavat clock mechanism. I was able to use a stepper motor to do all this reliably, so I am at a point where I need to decide which way to proceed. I like the Lavat method, as clocks are cheap to buy and hack.

As long as you provide it power

do you have a "radio controlled clock" or a so-called "Atomic clock"?

actually, the current direction reverses to effect the stepping action. The IC that is typically sealed in a dab of epoxy does contain an h-bridge. you can check out NXP.com PCA200x OR hackaday.com/2018/06/07/dive-inside-this-old-quartz-watch/ (or both, if you're really interested). The clever bit in the integrated poles is the asymmetry that results in a preferred clockwise rotation, as you mentioned in your comment.