The house may feel free to impeach whenever they want. But be aware that if they do, and the Senate has a trial and votes not to remove, then none of that evidence or criminal accusation can be brought up again if they impeach again for further crimes. The WH is certain to invoke double jeopardy rules and take that to the SC (I don’t think that has been legally tested before).
Maybe the Dems plan is to wait for ALL the evidence and investigation to be done, and then put it all in front of the Senate at once, instead of an impeachment per month. If the investigations continue to be drawn out or expanded, there are certainly downsides, e.g. 4 yrs of free action for presidenting instead of orange jumpsuit. Because you know a Republican House would love to hamstring a Dem WH with an impeachment every month forever.
But it’s not a court of law, so no double jeopardy exists. You can impeach and remove the president for almost anything the Senate/house likes.
So, the vote to impeach now, fail, and then wait a few days or week, and decide to impeach him again (maybe because he hates America), there is no reason why the evidence can’t be used again.
Now, you might loss the public, but that is something very different, but nothing in the rules say that you have to guilty of a crime to be impeached.
Sure, that isn’t in the Constitution. But the process is designed to mirror a civilian court Indictment (Impeachment vote in the House) and Trial (in the Senate), and the WH will absolutely argue the accepted legal standard of double jeopardy attaching when the Senate trial starts. How do you think this SC will rule on such a question.
Given the makeup of this SC, I think they’d actually say the Constitution doesn’t say anything about double jeopardy, so Congress can do whatever it wants.