The Senate isn't even a branch, it is a chamber of the bicameral Congress. It cannot speak for Congress alone, except within its plenary power of advice and consent, and treaty ratification.
Regarding impeachment, the Senate already has the consent of the House, but it still cannot compel the Chief Justice to appear (the Constitution would, if conditions apply). If the Senate disapproves of the Chief Justice's behavior, it can always seek the House to level impeachment charges against the Chief Justice for treason, bribery, or other high crimes and misdemeanors, or failure to demonstrate Article III "good behavior." On the latter point, "good behavior" does not mean refusal to cave to the demands of Congress, as much as Pelosi and Schumer wish it to be.
It appears that the Chief Justice took the literal meaning of the Constitution and decided that the President is not being tried and therefore his appearance is not called for. It is his call to make, and he made it. Based on that, citizen Trump should appeal to the Supreme Court to rule that the trial is unconstitutional because there is no penalty; he has already been removed from office by the Electoral College.
-PJ
Apparently, he is saying he doesn’t have the authority as per Constitution as he interprets it
But the Senate could come back and designate him giving him the authority
Then it would be back to him to refuse.
I would presume the Senate would normally not do that if they knew Roberts would refuse..but these are not normal times.
The president of the Senate is the Vice President. The Constitution specifies that the VP does not preside over the trial of a President because the VP has an obvious conflict of interest. But since Trump is a former President, I don't see why Kamala is not going to preside over the trial.