IIUC, the provision in the constitution for the legislature to reign in the executive is to refuse to pass any appropriation bills. ANY. Because the executive was known to intentionally pervert the intention of any bill that was passed.
Of course, that means that the feds would shut down. Originally that was much less of a big deal. These days...there aren't many places to homestead anymore. But it means shutting down Social security, the treasury, the military, the TSA, the Air traffic controllers, the FCC, etc. Somehow people don't seem to be quite willing to do that. The Feds have taken over many jobs that were intended to be done by the states. And they've used legal interpretations to weasel their way into all sorts of nooks and crannies where dislodging them would grossly inconvenience or endanger people. Because of this, shutting down the government would not be looked on kindly by the vast majority of citizens. Some of this was done with good intentions, but good intentions or malicious, it was done, and no other organization exists to do many of the jobs that need doing.
FWIW, congress is allowed to define whatever it chooses as "grounds for impeachment". If I recall correctly the only definition is "High crimes and misdemeanors", and that term is not further defined. But your point about "If you don't have the votes for legislation, you sure aren't going to have them for impeachment." is quite valid.
I do agree that the case is just grandstanding, though. IIRC the president can decide that he refuses to allow himself to be sued. (Otherwise every president would end up spending all his time defending himself against claims of impropriety...even before he took office.)