Indeed, those are the current Senate rules. The Senate has the opportunity to change or affirm the rules at the beginning of each session, and has, under extraordinary circumstances, the option of changing the rules in the middle of a session. The optics would be horrible, but they can change the rules in January when convening for the second session of the 116th Congress. They could adjourn the current session early, in case the House tries to bring articles to the Senate before the end of the year.
I'm not saying this is likely, but if Mitch doesn't care about the optics, he can pull some shenanigans to stall the trial. Historically, Senate trials don't start the day the House submits articles anyway. The articles are acknowledged, and then the tedious business of setting up a schedule begins. Witnesses must be identified and dragged in; absent Senators recalled (or present ones make up emergencies to get called away); the Chief Justice must arrange his schedule, the House needs time to prepare their charges, etc.
It's not instantaneous (and hasn't been in the past), despite the wording of the Senate rules.