Here is the schematic for the Bonus Unit Step Up solenoid:
Nip It Bonus Step Up (resized).jpg
The three switches on the right side of the schematic should deliver a single pulse during reset to advance the Bonus Unit to the 1000 point position. If you're getting a weak pulse it could be that one or more of those three switches is dirty or not properly gapped. The most likely of those would be the switch on the Score Motor at position 10A. If it's just barely making contact when it closes that might explain a brief pulse to the Bonus Unit solenoid.

You'll find a color chart on your schematic to help identify the wire colors used by each switch, and there's information about identifying Bally Score Motor switches here: https://pinside.com/pinball/forum/topic/sky-kings-power-free-play-and-top-hole-issue#post-4457680

/Mark

The stepper has to mechanically work well before the circuitry can be expected to control it properly. If you turn off the power you can test the stepper by manually pushing in the solenoid plungers and releasing them (one at a time). The spring tension that returns the plungers to their rest positions should be enough to increment or decrement the stepper by one step. If you can't reliably get the stepper to increment and decrement across its entire range by just pushing in the plungers you probably have grease, grime, friction and/or wear to resolve.

There are three different circuits that can reset the Bonus Unit:
Nip It Bonus Unit Reset relay (resized).jpg
One experiment you can try to isolate the problem is to set the Bonus Unit at a high bonus (e.g. 8000 as you mentioned) and then reset or tilt the game. Those two scenarios use the left and center legs of the schematic. If they can reliably reset the Bonus Unit the problem is more likely in the right leg than in the Score Motor switch or Bonus Unit itself. Conversely, if all three resetting mechanisms have the same problem, you're more likely to have an issue with the Score Motor switch or the Bonus Unit.

A voltage change will have no meaningful effect on the speed of the score motor. The voltage does affect the strength, or the torque of the motor, but not its speed. (technical explanation below *)

A more likely explanation for the speed of the two motors is that one is geared for 50Hz AC power and the other is geared for 60Hz power. You could check the motors themselves. They're often stamped with spec numbers of various kinds.

The Ball Grabber solenoid (which is wired to the mains BTW so be very careful) is controlled by the Ball Grabber relay. I thought that perhaps they would have used a flipper solenoid with two windings for the Ball Grabber but they didn't. The grabber button fires the Ball Grabber relay which stays on until the Ball Grabber reaches its End of Stroke switch. However if the player holds the Ball Grabber button in, both the relay and solenoid will remain on indefinitely.

(*) Score motors are built with shaded pole induction motors which are cheap and reliable but also inefficient. The AC voltage applied to them creates a rotating magnetic field and the rotor tracks that magnetic field as it spins. Changing the voltage will make those magnetic fields stronger or weaker but they still rotate at the same frequency based on the frequency of the voltage (e.g. 50 or 60Hz) so the rotor speed doesn't change unless the voltage (and magnetic field strength) is so low that it can no longer spin the rotor at all.