I agree that this 200+ year old document could stand an overhaul. But please look at the second requirement. "When sitting for that Purpose [the purpose of trying an impeached person], they shall be on Oath or Affirmation."
Why this requirement? The Senators are already bound by oath or affirmation to function in support of the Constitution. It would indeed be reasonable to say this requirement removes them temporarily from their normal duties as Senators and places them into the fact-determining part of the trial. There is no reason to doubt that the founding fathers envisioned anything other than that the trial would proceed according to established court procedure. I agree, the Chief Justice does not make rules for a trial, they were then in existence and may have been amended according to law over the years. But they do exist. It is the duty of the presiding judge to see that proper trial rules are followed.
Please note that the impeachment itself does not require a special oath or affirmation. The impeachment is merely an allegation of an wrongdoing for which the wrongdoer can be removed from Office if the allegation is found to be true. With respect to the trial, in contrast, it is of utmost importance that established rules for the Judiciary be followed for justice to prevail. Isn't it the purpose of this second requirement to bind the senators to their separate and distinct duty of determining the facts and letting the presiding judge make determinations on questions of law?
If indeed the Majority were allowed to make rules to deny facts and testimony to be brought before that trial, would the undertaking not become a farce? What would happen to our democracy?