House speaker position if the Dems lose their majority

You’ve cited part of the Constitution relating to the conduct of the Senate, but not all of it. In addition to the provision making the Vice-President the President of the Senate and providing for the president pro tem (Article I, s. 3, clauses 4 & 5), there is also the provision that the Senate sets its own rules of procedure (Article 1, s. 5, clause 2):

Thus, the President of the Senate is not a dictator and doesn’t get to set the rules of procedure. It’s the Senate as a whole which sets the rules. It has done so, and sets out the process for points of order, in Rule XX of the Senate Rules:

A paper from the Congressional Research Office explains in more detail how the appeal process works: Points of Order, Rulings, and Appeals in the Senate.

Thus, the Presiding Officer is a relatively weak one, as any ruling made by the Presiding Officer can be appealed immediately to the Senate as a whole, and the Senate can overrule the ruling of the Presiding Officer.

The relative weakness of the Presiding Officer is a point made in the article which you cited. The Senate has chosen to make the officer a weak position, unlike the Speaker of the House, because the Vice-President can at any time assume the chair. Given the close connection between the President and the Vice-President, there is the concern that giving the Presiding Officer a strong role would indirectly give the President the power to direct the debates and legislative agenda in the Senate, exactly as you are suggesting.

The article you cite also explains the origin of the rule that the Presiding Officer always first recognises the Majority Leader:

The article is correct that this practice is not set out in the Rules of the Senate, but is rather a practice established by the President of the Senate in 1937. Yes, a new Vice President could try to change that rule - but it is the Senate, not the Vice President, who has the final say.

If the Vice President failed to recognise the Majority Leader, any Senator could rise on a Question of Order and contest the failure of the Vice President to follow the practice established in 1937. There would be debate, and the Senate as a whole would then vote whether to uphold the Question of Order raised by the Senator and whether to overrule the Presiding Officer. The Vice-President would not have a vote on that matter, unless the Senate is tied.

If the Senate allows the appeal, the Presiding Officer is overruled and must recognise the Majority Leader according to the 1937 practice.

If the Republicans hold the majority in the Senate, can you see any chance where they would reject the appeal, uphold the ruling from Vice-President Harris, and effectively cede control of the Senate to the Vice-President?

(And, apologies to @UltraVires - I see that he was the one who made the point that there could be an appeal to the Senate on a question of order, not D’Anconia. Late night posting sometimes leads to errors. :thinking: )