So, first let's get rid of the idea of "simulating a motion toward home." That seems to run together two different ideas: simulating a pitch, which is illegal when F1 is not legally engaged, and moving toward home, which is illegal if F1 subsequently fails to pitch. Quite different infractions that need to be kept distinct: one is done off the rubber, the other is done on it.
Second, the set position is defined by the feet, not the hands. "Coming set" is another thing altogether, and is a moment during a pitch from the set position. Precision about the pitching terminology — which is not common among players — will help you get all this right.
Next, back to basics: from the set, F1 may legally do 3 things. He may disengage, pitch to the batter, or step and throw/feint to a base. This is true for both RHP and LHP: the rules make absolutely no distinction between what F1 may do based on his pitching hand.
The step required by "step and throw" has a fairly precise definition: the free foot must gain distance and direction toward the base where the throw is going. Distance: the free foot must move out of its footprint to count as a step. Direction: that motion must be approximately toward the base (within 45° either way).
These are the only restrictions on the step. You'll notice they do not mention the pivot foot.
So, finally: we all start umpiring from somewhere. You played the game, and you know it from that experience. That's fine, no problem at all. According to your experience, a RHP will almost always use a jump/jab step for a pickoff to 1B. That's legal. Then this question comes up: the mistake is to assume that because you've seen something done, that defines what's legal. The RULES define what's legal; if you're not certain (and you shouldn't be), look it up.
Particularly for pitching rules, umpires need to get their noses in the book more. Too many assumptions about what's legal based on what's common.