They're in a pretty tough spot. We all know how an actual impeachment proceeding will go. The GOP could have video of Trump stabbing a baby and they wouldn't vote to convict Trump in sufficient numbers in the Senate. They'd claim fake news, Fox News would say the video is doctored, the ultra right would say the baby had it coming, etc. So he's not going to be successfully removed from office via impeachment.
Option 1: Hold the vote in the house, it passes, goes to Senate and fails.
Option 2: Drop the whole thing.
Either way, Trump will proclaim he has been "totally exonerated," and if there's a trial a chunk of the population will buy it because "acquitted by Senate" will appear in the headlines. Regardless of what evidence was detailed at trial. We tend to forget on a message board like this that a large portion of the population just doesn't give a **** and barely pays attention to politics. So the hope is that the evidence unveiled will convince enough Democrats to get fired up to vote out the criminal president, and enough Republicans will stay home in 2020.
Problem is, you run the risk of firing up Republicans who try and close ranks around their guy, and turning off Democrats who see the Democrats failing in their goal. The question becomes about how many of each category you're going to affect.
Personally, I think doing nothing is the worse option. It may be personal bias, because if the Democrats walk away and don't even try to remove a criminal from the white house, I'm not terribly motivated to go out and vote for the useless ****s. You have to try and do the right thing, even if you know you will fail.