From the Constitution, Section 3, fourth paragraph:
The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.
The Senate shall chuse their other Officers, and also a President pro tempore, in the absence of the Vice President, or when he shall exercise the Office of President of the United States.
I did a few searches on the meaning of the word president and its meaning has not changed since 1788.
And just as importantly, maybe even more so, is this section from Article VI, second paragraph:
This Constitution, … shall be the supreme Law of the Land; ….
The Constitution is “superior” to all laws made by Congress. It is superior to all rules, regulations, and traditions implemented by the Senate. If there is ANY disagreement, anyone who might take issue with these ideas should be suggested/requested/even required, to refer to the Constitution for clarification.
From my reading, Harris could establish positive control and stop any filibuster before it began.
However, this might be a problem with committees. This action will indeed upset apple-carts, and ruffle feathers, to the point to outright plucking. But maybe this is something we need.