For the benefit of anyone looking for the answer:
No. No player gets priority during the untap step; the first time players receive priority in a turn is in the upkeep. By the time players receive priority, abilities that trigger at the beginning of the upkeep have already triggered and been placed onto the stack. And lastly, because an ability on the stack exists independently of its source, destroying or otherwise removing the source will not counter or remove the ability. The abilities will still resolve unless you counter them directly or otherwise remove them from the stack.