This is definitely not how it was “designed”. No one intended the president to obstruct any effort at congressional oversight. It was not designed to allow the president to stall for years to avoid accountability.
Then make the courts work faster. They are there to ensure that the law is followed, and attempting to remove a duly elected president should NOT be done rapidly. If the judges feel that he is using the courts frivolously, they can reject his filings out of hand. If they, however, choose to hear the arguments and render their rulings in due time, it is their prerogative to do so. We do not want to do something like this in the heat of the moment.
Let's put it this way. Just how easy would you want removing a president to be? If the democrats succeed in removing Trump without the full concurrence of the judicial branch, and manage to win the White House (I know, it has to happen some time), how easy to you want it to be for the Republicans to find some questionable activity and ram through an impeachment of their own? You would be setting up a scenario where the only way a president could be sure of a full term is to have a majority in the House.
Anyway, it's all a moot point. The democrats haven't made a very strong case to get rid of Trump, and the further they push it like this, the worse it will be for them.