It sounds very good, and I wish it were true. Unfortunately, it is not.
TL;DR: No, Kamala Harris will not be able to take control of the Senate's agenda away from McConnell if Democrats don't win both Georgia runoffs. A simply majority vote of the Senate determines how it operates and if she tries, the GOP majority will simply change the rules to formalize the informal policies that give McConnell his power.
The Constitution states that "each [chamber of Congress] may determine the Rules of its Proceedings."
Senate Rule XX (often referred to as the "nuclear option") reads, in applicable part:
A question of order may be raised at any stage of the proceedings...[such a question] shall be decided by the Presiding Officer without debate, subject to an appeal to the Senate...and every appeal therefrom shall be decided at once, and without debate.
Rule XX means that no matter what the Rules of the Senate may say or what the President of the Senate may want to do, a simple majority of the Senate can always change the current language of a Rule into whatever the majority wants the Rule to be.
In November, 2013, we saw Rule XX in action when Senator Harry Reid used a simple majority to change the Rules so that McConnell's minority could not "filibuster" President Obama's nominees (other than the Supreme Court). The exchange on the floor went like this:
Mr. REID. I raise a point of order that the vote on cloture under rule XXII for all nominations other than for the Supreme Court of the United States is by majority vote.
[The phrasing is complicated, but this is Senator Reid stating the new rule he would like to put into effect.]
The PRESIDENT pro tempore. Under the rules, the point of order is not sustained.
[The chair, after consultation with the parliamentarian, disagrees with Reid's "interpretation" of the Rule as it now stands.]
Mr. REID. I appeal the ruling of the Chair and ask for the yeas and nays. (48–52 vote on upholding ruling of the chair)
[Reid appeals the ruling, though he knows that the chair is actually interpreting the rule as written and wins his appeal 52-48 (the chair loses the vote 48-52).]
The PRESIDENT pro tempore. The decision of the Chair is not sustained.
The PRESIDENT pro tempore. Under the precedent set by the Senate today, November 21, 2013, the threshold for cloture on nominations, not including those to the Supreme Court of the United States, is now a majority. That is the ruling of the Chair. (Congressional Record of November 21, 2013)
And thus, the threshold for cloture votes was changed; the simply majority of vote of the Senate is binding.
I provide the text from 2013 of Rule XX in action so that I can now illustrate, using almost identical procedure, what would happen if a future Vice President Kamala Harris were to decide not to abide by the convention of "Priority Recognition," established in 1937 by Vice President Nance.
The moment Vice President Harris (who would be "The PRESIDENT" in the Record), started to recognize someone other than Sen. McConnell at the start of the day's business, the following (hypothetical) exchange would occur:
The PRESIDENT. I recognize [anyone who is not McConnell].
Mr. MCCONNELL. I raise a point of order that under rule XIX(1)(a) the Majority Leader, who shall be the Senator chosen by the political party with which a majority of the members of the Senate affiliate, must be recognized by the Presiding Officer first.
The PRESIDENT. Under the rules, the point of order is not sustained.
Mr. MCCONNELL. I appeal the ruling of the Chair and ask for the yeas and nays. (48–52 vote on upholding ruling of the chair, Harris loses)
The PRESIDENT. The decision of the Chair is not sustained.
The PRESIDENT. Under the precedent set by the Senate today, January 21, 2021, the Presiding Officer shall first recognize the Majority Leader. That is the ruling of the Chair.
And just that fast, Vice President Nance's informal policy would become a binding precedent of the Senate and the meaning of Rule XIX(1)(a) would be changed.
There would be nothing that President of the Senate Harris would be able to do about it.
This is why, in the 83 years since Vice President Nance adopted this informal rule, no Vice President affiliated with the minority party of the Senate has attempted the move that folks spreading this argument seem to think Harris could just walk in and do on January 21. The rules currently support her ability to do so, it's just that the rules would be changed in one vote.
For better or worse, the Senate governs itself by majority rule and the Constitution makes the Vice President the Presiding Officer, not dictator of the Senate.
This "hardball" tactic will not be employed; not because Democrats can't play hardball, but because it would fail immediately.