> In general, the Senate's ability to set its own rules surely cannot be unrestricted. For example, when the Democrats had 60 votes in the Senate would it have been constitutional for them to create a new rule that all tax decreases require unanimous consent of the Senate? Or for Republicans in the same situation to create a rule that all tax increases require unanimous consent? Both of these changes would seem to be within the rule-making authority of the Senate. However both rules would be profoundly undemocratic because they would generally prevent a majority (or supermajority) of US voters from changing US tax law in the future.
I think we should clarify that there is no "filibuster rule" in the Senate. The rule is that debate by default is unlimited; in order to close debate on any non-rule change, sixty senators must agree to invoke "cloture," after which debate continues for up to thirty hours under various restrictions. Then the vote on the underlying measure (e.g. a bill, a resolution, advice & consent on an executive matter) takes place with the majority vote rule. (Debate about rule changes can only be ended by 2/3 of the Senate, followed by a majority vote on the rule change itself.)
With that settled, we should also consider that the Constitution itself mandates super-majorities in the Senate for various reasons, including expelling a senator; giving advice and consent for the president to ratify a treaty; convicting an impeached president, vice president, other officer or judge; overriding a veto; or proposing an amendment to the Constitution. It further specifies certain procedures that can be invoked by less than a majority, like the 1/5 vote to record the names of those voting on any given measure, or the minority vote required to compel the presence of absent senators in the absence of a quorum. The Constitution clearly contemplates these situations where the vice president's vote wouldn't matter, so it stands to reason that the Senate is free to adopt rules requiring a different vote threshold than simple majority.
As to the hypothetical about rules governing tax legislation, I see nothing in the Constitution forbidding such a ludicrous procedure. I do see loads of evidence that it would be extremely impractical, including the 2/3 majority required to invoke couture on rule amendments; the "nuclear option" precedent; and the fact that the tit-for-tat is possible.
Finally, all legislative bodies have substantially undemocratic procedures. Referring matters to committee is undemocratic, restricting floor amendments is undemocratic, points of order are undemocratic, etc. It seems to me that preserving the minority's right to further debate is not so different.