They absolutely can proceed as described. You are confusing House procedure with Constitutional requirements. The House procedure doesn’t take precedent over Senate procedure. The Constitution is silent on the impeachment process beyond giving the House the power to impeach and the Senate the power to conduct the trial. There is no Constitutional requirement that there be impeachment managers/prosecutors or that the articles be formally transmitted to the Senate. The House has impeached President Trump. That’s it. It’s Constitutional function is at an end. Any further involvement is a matter of negotiation with the Senate. Additionally, the Senate could easily adopt a rule to the effect that the House passing articles of impeachment triggers the process for a trial in the Senate with no further House action. There is no requirement that the Senate wait until the House appoints “impeachment managers/prosecutors” There is certainly no obligation by the Senate to allow those “managers” to address the Senate or even have access to the Senate. In fact, Mitch McConnell has exactly as much authority to try this impeachment without any (name removed by moderator)ut from the House as the House has to refuse to send the Articles of Impeachment to the Senate. In short, Pelosi’s gambit is, well, toothless. It has NO legal power.
Trump got impeached for what Pelosi is now doing: Abuse of Power (extortion, bribery, Quid Pro Quo) and Obstruction of Congress (refusing the President his right to a trial for possible acquittal by withholding impeachment records from the Senate).