Harris is not going to limit her role to sitting around the Capitol to preside whenever the Senate is in session.
And to the extent she chooses to abandon the tradition of giving the majority leader priority when it comes to recognizing a Senator to be heard, it will be simple for the Republicans, if they're in the majority, to change the rules of the Senate to require the presiding officer to give priority to the majority leader.
This could be done if she's not present by having McConnell raise a point of order suggesting that, notwithstanding the rule that says the presiding officer must recognize the first person to seek recognition, the majority leader gets priority. The friendly presiding officer denies the point of order and McConnell appeals the ruling and it merely takes a majority vote to overrule the presiding officer and change the rule (i.e., the "nuclear option"
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Indeed, even if Harris was in the chair, the same scenario could and would play out. McConnell would raise a point of order, Harris would deny it, and a majority of the senate would overrule her.
So this idea is a non-starter.
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