In the pits, from a stop, especially including a start anything other than perfectly straight, the bike isn't totally stable yet. Any motion, including that of putting the feet up, can be upsetting to the bike, so they don't. Don't look for too much there because there isn't much. It's just easy and no advantage to do differently.
On a race start, that's different. And get the idea that they push off
from one side or the other out of your head because that's not at all what's going on. Standing still, the bike is balanced with both feet doing nothing more than holding the bike from falling. The weight on each foot just before launch is likely measured in ounces, not more than a pound per foot, and near to perfectly equal weight on each foot, holding the bike straight up, balanced, so when it's time to go the bike launches straight. And there is no pushing off involved, just using the bike's power. For those first few yards you can't afford any upset, there is no reason to put the feet up and plenty of reason not to, not wanting to upset anything. Also, the legs down has the center of weight lowered which further helps keep the front down, a freebie. The total focus is on the clutch and throttle keeping accelleration maxed out, the bike straight, and the front wheel reasonably close to the ground, ideally no more than an inch or two up. The higher you allow the front wheel to come up the higher the center of weight is, and the less accelleration it will stand without flipping. All of the focus is there, none left as well as none needed or wanted to move limbs, so they don't. As the speed picks up and the bike stabilizes, is less prone to wheelie and will be getting ready for the first shift, and the feet start to smoothly move up to the pegs, all ready and just in time.
Hope that adds some understanding.