I'm convinced none of this is about impeachment and actual removal but rather the House Dems trying to give their weak ass candidate pool winner a card to play during the campaign. The candidates they have are not going to get elected on the basis of their policy positions or electability so they need some help in the form of running against an "impeached" president. The media of course plays a big role in pushing this.
This is their Hail Mary which is why they don't want it in the open because many would see their "witnesses" get shot full of holes and their impeachment plan falls apart. They weren't counting on Trump releasing the transcript but it doesn't seem to have slowed them down in their whistleblower narrative. They need to get this to the Senate. As a side benefit for many House Dems, it helps them with their base given many of them ran on impeaching Trump.
And then in the Senate, they also want it to appear bi-partisan so they are seeking the GJ info from the Mueller report to give a few squishy Rs in the Senate a little more reason to vote for removal. Current Senate makeup is 51 R's, 47 D's, and 2 I's who caucus with the Dems. So basically 51-49. They won't get to the required number for actual removal but it helps their "narrative" if 2-3 Rs flip on Trump and vote to remove so they have a majority number voting for removal.
The idea at the end of the day is that every time the word Trump is mentioned during the 2020 campaign, the media and Dems will always associate it with "impeached and bipartisan majority voted for removal".