The United States Senate is currently (August 2021) split 50-50 between Democrats and Republicans. As such the Vice President gets the deciding vote:
The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.
(Constitution, Article I Section 3)
If the current president ceases to be president for whatever reason (death, resignation, impeachment, etc.) the current vice president would become president:
In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President,
(Constitution, Article II Section 1)
If that happens, a new vice president would have to be confirmed by a majority vote of both houses of Congress:
Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.
(25th Amendment, Section 2)
It seems fairly likely that any nominee for the vacant vice presidency would be a Democrat. Thus confirming the nominee would give the Democrats back the tie-breaking vote in the Senate. But for the vote itself on the vice presidential nominee there would be no tie-breaker vote since the Senate is split 50-50 and there is no vice president (as per this question and this question). So Republicans could simply refuse to confirm a vice president, thus keeping the Senate without a tie-breaking vote indefinitely (until the next election). This could mean that the Senate might be able to get nothing done, if every vote is split along party lines.
This seems like an odd situation. Are any of my premises above incorrect? Or is there any other way that Republicans could be prevented from doing this, to ensure continuity of the Senate's operations? And if they could do it, would there be any reason for them not to do it?