Setting aside the detail errors (6a has no application, not a criminal trial; "mandamus" is the name of a judicial writ), what you propose is an inversion of the usual process.
I won't say it can't be done, because everything in the way this case is proceeding is unprecedented.
Anyway, what you are suggesting is that the accused, the defendant, present himself to the court sans accusation and accuser. Would the senate entertain that sort of move? There is no precedent, no procedure to do this.
But a senate committee could take up Trump's defense and hold hearings, call witnesses, and so on. That creates a record in the Senate.
Right now, I'm of a mind to let this fester. I think it hurts Nancy and the DEMs more than it hurts Trump. A number of DEMs are sensing that the public is puzzled at best by Nancy's maneuver.
I checked all the precedents, and what Pelosi is doing is truly unprecedented. The house might wait a few days to name managers, but it always plays the roles designed into the impeachment process, including taking the initiative to take its case to the senate.
apparently henry hyde presented the articles on the same day (a saturday) to the senate in 1998. in 1868 it apparently happened over a 2 day period.
(so much for precedents...)