They key is the Constitution left HOW the two houses of the legislature up to whatever rules and procedures the houses separately make for themselves. What is being discussed has zero to do with Constitutional law regarding the legislature, and is totally about “accepted traditions”.
If it is an accepted tradition that the presiding officer of the Senate, which happens to be the Vice President of the United States, CAN over rule the parliamentarian, opponents can squeal, but in fact no laws or rules would be “broken”, just “tradition” terribly offended.
I am a mugwump on the issue. One part of me would like the immediate result, and the other part sees how we could come to hate this breaking of the traditions, if and when the Dims had the White House and 50+% majorities in both houses.
I often wondered what would happen if the VP took the job as president of the Senate seriously. He could actually be the tough guy more so than the Majority leader. Would be fun to watch but we would need a Patton or MacArthur as VP.