Why would voting no on legislation to allow for it to be reintroduced be beneficial?

Question

The US Senate introduced a pair of bills which failed to get past the filibuster 60-vote threshold.

In this article and this one, it was indicated that the Senate Majority Leader, changed his vote from Yes to No and by doing so it gave him the option to bring the bill back up again in the future.

The Rules of the Senate seems to indicate that senators are called to vote in alphabetical order and cannot change their votes once made per Section XII:

When the yeas and nays are ordered, the names of Senators shall be called alphabetically; and each Senator shall, without debate, declare his assent or dissent to the question, unless excused by the Senate; and no Senator shall be permitted to vote after the decision shall have been announced by the Presiding Officer, but may for sufficient reasons, with unanimous consent, change or withdraw his vote. No motion to suspend this rule shall be in order, nor shall the Presiding Officer entertain any request to suspend it by unanimous consent.

Furthermore, under Section XIII of the rules, it appears to allow for reintroduction of a bill:

When a question has been decided by the Senate, any Senator voting with the prevailing side or who has not voted may, on the same day or on either of the next two days of actual session thereafter, move a reconsideration...

I think this means that if a Senator wants to reintroduce legislation for debate after it's been voted upon to not introduce it, they may do so, but only within 2 session days.

My questions are as follows:

1. Why is the Senate Majority Leader allowed to change his vote at all, it seems unlikely they would receive unanimous consent? Is this a privilege of the Majority Leader?
2. Am I understanding the rule correctly regarding voting No? Is a senator only allowed to reintroduce the legislation within 2 session days? If so, it doesn't seem very advantageous. Or am I misreading it and there's some loophole that allows it to be reintroduced at any time during the current session?

Answer 1

Any member of congress can change their vote as many times as they want as long as the vote is still open.

The Rules of the Senate seems to indicate that senators are called to vote in alphabetical order and cannot change their votes once made. Furthermore, under Section XIII of the rules, it appears to allow for reintroduction of a bill:

This is a special type of voting and normally they vote by electronic means and even with a voice vote members are still allowed to change their vote before the vote ends.

AS for the second question I think you are misunderstanding what was being voted on. The vote was to end the filibuster which is required in order for the bill itself to come up for a vote. In this case the rules say you have to be on the winning side to attempt to bring it up again.

When a question has been decided by the Senate, any Senator voting with the prevailing side or who has not voted may, on the same day or on either of the next two days of actual session thereafter, move a reconsideration; and if the Senate shall refuse to reconsider such a motion entered, or if such a motion is withdrawn by leave of the Senate, or if upon reconsideration the Senate shall affirm its first decision, no further motion to reconsider shall be in order unless by unanimous consent. Every motion to reconsider shall be decided by a majority vote, and may be laid on the table without affecting the question in reference to which the same is made, which shall be a final disposition of the motion.

They change their votes to no once it is clear that the vote to end the filibuster won't succeed so that they can attempt to bring it up again. This isn't a loophole but rather preventing someone from brining a failing motion up for vote over and over again.