No, what I'm saying is the Constitution gives the Senate the sole right to try the impeachment. Period.
Putting arbitrary restrictions on how they conduct that trial, like barring new evidence, may be fair, reasonable or defensible but it isn't Constitutional.
They judge the case as presented, they do not build both sides of a case. The trial consists of what was presented, not what they create.
You are missing the point. The Senate can make it's own rules as to how it "judges" the content of that which the House presents in their article of impeachment. The Senate is constitutionally barred from helping the house put forth it's case or assist the House in the discovery phase. "New" evidence or testimony falls under the discovery phase of an impeachment, NOT the judging phase. The Senate judges whatever the House discovers and argues.