The Senate doesn’t have “authority” over POTUS. They’re co-equal. They do have a check on abuse of power, which is why the legal professors the Democrats brought forth were so intent on defining that term.I think the dynamics of witnesses and Trump trying to exert executive privilege will be fascinating. Senate calls a witness and Trump claims total immunity. The mere act of Roberts signing off on a subpoena would effectively render the total immunity argument moot. So the witness shows up for questions, but then asserts actual privilege that has some precedent in court rulings (like direct comms with POTUS).
House Managers will challenge that, arguing and citing the few cases on the subject, where privilege is always balanced against the public's right to transparent government. If there's any proceeding in which privilege goes to zero, it would be an impeachment trial. Roberts would then have full authority to make that determination.
However, any ruling he makes could be overturned with a simple 51 vote majority. Because of that detail, I suspect Roberts is generally going to give deference to the Senate's authority over POTUS. Constitutionally this makes sense, sense the "sole power" to try impeachments is vested in the Senate.
If there's any tiny chance of Trump being removed, it would probably be for efforts to obstruct the trial itself. That could turn personal real quick.
Congress is welcome to challenge Trump’s assertions in court. Why haven’t they done so with any seriousness yet? Could it be because they don’t really want to narrow the application of Executive privilege. Remember that the door swings both ways on that. President Obama claimed Executive privilege plenty and openly dared Republicans to do something about it. Every President in the future will as well, especially if the precedent of non-stop attacks toward Presidents from Congress continues into future Presidents’ terms.