Interesting question. You have a choice as to who gets to start with an almost complete stroke and who starts with a bit less than half a stroke. Starting OOP has that different feel to begin with, but you've been through that.

One thought: when I ride with my son on the back (he's bigger and stronger than I, but not so much bigger he can't go on the back, so this is the preferred configuration), I am holding the bike more by either leaning the top tube (ever so slightly) against one leg, or by practically sitting on the top tube. So it's not the arms and shoulders that are steadying the bike. It needs to be slightly tilted and pressing against the on-ground leg when I start up, and then in the single motion that gets my seat onto the saddle and the high pedal to the low position we get going. This motion is brief and it is during that motion that we have the initial acceleration that is adequate to transition from holding the bike up to its balancing on its own. The arms/hands are only needed to hold the brakes when stopped. But I haven't tried OOP with my son yet (he's away at school).