Wow, that is a lot of intensity over toothpaste! :)
I say as long as the toothpaste stays on the brush long enough to get to their mouths, you're good.
But since the question is already out there, why on earth are you having them intentionally touch the toothpaste tube to the toothbrush? That is pretty much the second-germiest thing in the bathroom! Giving the germs a free ride every day into the whole tube is not a good plan, I wouldn't think. Seems the "mess" you're focused on isn't where the real action is.
Besides, toothpaste has never been that tough for my little ones to get off their fingers. Maybe you can check the fingers at the same time you check the teeth; just make it part of the routine. Send them back for another quick rinse the same way you would send them back for a rebrush if there was still crud on the teeth or breath wasn't smelling quite clean enough yet!