In a parliamentary system, the Democrats would be clamoring for a vote of no confidence. And they would have every right to do so; that's the way parliamentary systems work. We don't have that mechanism, so the dems are trying to manufacture a crime. If they establish the precedent, future presidents are likely to face this routinely whenever the other party has control of the House.
Some of the talking heads should raise the obvious hypothetical. Suppose Trump is succeeded by a Democrat. On the way out the door, Trump can appoint a special counsel to investigate the new president concerning "all matters" pertaining to whatever seems timely, and to prosecute any violations of law that he might find. This would be an open ended fishing expedition. And any attempt to fight back, including appeals to the court of public opinion, would be deemed criminal "obstruction." Is that really the game we want to start playing?
You seem to forget that we’re talking about spineless Republicans. They would do no such thing to a Democrat president.