Question Regarding Sausage Making

Question: Why is everyone talking about the need for the US Senate to have 60 votes to pass any bill?

Here's my understanding:

• The US Senate requires a vote on a bill before it is officially passed by that body. In order to pass that bill, it is presented for an up-or-down vote on the floor of the Senate, and a simple majority of Yeas from all Senators* is all that is required for the bill to be passed. In our current case, if all 100 US Senators were present for a vote, only 51 Yeas would be required to pass that bill. In the event that there are only 50 Yeas, the Vice President of the United States can cast the 51st vote to pass the bill.
• As a procedural technique, the Senate permits a filibuster under Senate Rule 22. I believe the term "filibuster" is a fancy shorthand for "invoking an indeterminate period of debate in which the Senate may not vote on the bill in question, nor may it adjourn". Basically, "we can't vote while I'm still talking--even if I'm just babbling about the petunias I'm growing in my front yard". Interestingly, the Senate Majority Leader seems to have the power to require a "traditional filibuster", in which the filibuster-ers must speak continuously during their filibuster; gaps in coverage, as it were, permit a vote on the bill to take place.
• The only vote that may occur during a Senate filibuster is a cloture vote. I understand that "cloture" is short for "Okay, you stop talking now and we finish this business". In order for a cloture vote to pass--i.e., in order to stop a filibuster--a 3/5ths majority of the Senate must vote Yea. So, in our current circumstances, 60 Senators** must vote to end a filibuster.
• As a matter of fact, Senators are autonomous lawmakers. They have the right to vote Yea or Nay on any bill as befits their conscience or their perception of their constituents or their current mood or their astrological forecast or whatever. As a matter of practice, Senators vote with their caucus on procedural matters. So if a Senator wanted to vote Nay on a particular bill, but was told by his or her leader to vote Yea on cloture, he or she would indeed vote Yea on the cloture and Nay on the bill.
  

Now here's what I don't understand: barring any factual errors in the above, where O where is there a requirement that 60 Yea votes are required to pass any Senate bill? Or 65? Or 80? Why does a bill have to be constructed to attract 60 or more votes in the Senate?

As an example from the Big Game perspective: if I were the current Senate Majority Leader and I were faced with a bill I wanted passed out of the Senate (wherein I have, say, 55 Yea votes to count on and 59 other Senators caucusing with me), why wouldn't I just do the following:

1. Call the bill to a vote.
2. If the opposition to the bill decides to filibuster, permit it as I must. But require a traditional filibuster.
3. Watch for perhaps half a day as the opposition rants endlessly about "demonic welfare states" or reads aloud "The Joy of Cooking". Perhaps I'd also eat some popcorn, write some important emails, and rest secure in the knowledge that the C-SPAN footage of this spectacle will be used to excoriate these guys in the cable news that evening. Hey, perhaps this overt and ridiculous obstructionism would also win my team a couple of seats in the next two Senate elections.
4. When I'm sufficiently bored and/or tired, call for a cloture vote. The other 59 members of my caucus, of which (recall) I am the leader, have been previously informed of the following:
• Of course, vote however you like on the bill--that's your right. But you vote Yea to end a filibuster if I call for cloture.
• If you decide to vote Nay on cloture, there will be consequences. For example, all that support and PAC money you've enjoyed? Forget about it. You're not caucusing with us anymore if you can't get behind a simple procedural vote. Indeed, we'll be doubling the funding we're denying you and spending it on the person who would have been your most promising primary challenger. What do I mean by "would have been"? You see--you no longer get to use our letter by your name--you're not with us any longer. You may join the other guys if you like, or perhaps start up a third party. Those are really popular and effective, so I hear.
  

Is there a reason this isn't done? I still fail to see why Senate Democrats are currently bending over backwards to work around or obviate the Senate rules here, especially since it appears that they can work right through them to not only get bills passed, but perhaps also make some significant political gains.

My vote's "Yeah, whatever",
NFK
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*Is it "all Senators" or "all Senators in attendance"? For example, Senate business may proceed in the event that there is a quorum of 90 Senators present, but would that mean that a vote on a bill only requires 46 Yeas from Senators (or 45 Yeas + VPOTUS)?

**Similar question as in *. Is it a 3/5ths majority of all Senators, or just a 3/5ths majority of all Senators present?