The people of each state get to elect their representatives and senators, not people out of state who disagree with a particular state's outcome.
The Senate rules are just that, the Senate rules. Whatever the Senate decides is how it operates, the Supreme Court, in an Obama case, decided it is for the Senate to decide how it's rules operate, and not he executive branch.
Just what power being President of the Senate gives the Vice-President is pure conjecture, it hasn't been used much. When Washington went over to the Senate, for some advice and consent, he was told to leave, that was their jurisdiction, not his. Harris would face the same fate, even if the rules have to be changed.
Find McConnell no worse, just different, than Harry Reid, both political creatures, with little care for how their perceived.