It's more like the president is out for at most 21 days. The 21-day limit only applies when no vote is taken. If a vote is taken, and fails to get 2/3 in both houses, the entire process completes in the president's favor. Done. You could in theory keep the president out of power simply by failing to have a vote (the "deliberately dropped packet" approach), but there's no provision for voting over and over again on the same question without new cause. You can't just make up such a provision that would be an exception to how anything else in congress (e.g. impeachment) works.