interesting take
here's what i read, seems a little at odds with your statement
The Constitution requires senators to take a special oath or affirmation to participate in impeachment proceedings. It doesn’t specify what the oath must say, though the chamber’s practice has been to require each senator to promise “impartial justice.”
The wording of the oath was established in the first impeachment proceedings, the 1798 trial of Senator William Blount: “I, (name) solemnly swear, (or affirm, as the case may be,) that in all things appertaining to the trial of the impeachment of (name), I will do impartial justice, according to law.”
Yes "Appertaining to the trial" they must follow the rules of the senate trial
Since the trial rules are formed by the Senate and may differ from any previous trial...the oath to adhere to these rules impartially, must be reinstated at each seperate trial impartial and ilregardless as to political affirmation
All parties must follow the same trial rules set forth by the senate as an entirety