There is no constitutional basis for the filibuster. It's just a rule the Senate invented. And they can un-invent it. Whenever the new Congress forms, the Senate Majority Leader sets the rules for that Congress. This includes the number of seats on committees and so forth.
What you have is Democrats who hide behind "institutions" as an excuse to do nothing and Republicans who quite literally will change the rules whenever it suits them. Examples include:
1. Firing the Senate parliamentarian when he blocked the Senate from advancing Bush tax cuts;
2. Deciding completely without basis that with 11 months left of Obama's term, it somehow wasn't proper to hold a hearing or a vote on Garland's nomination to the Supreme Court; and
3. Deciding that any nomination (not just judicial) couldn't be filibustered (Harry Reid had previously changed the rules to make Supreme Court nominations filibuster-proof).
But what is the filibuster? Up until I believe the 1970s it was a process where a Senator who had the floor could refuse to yield and keep talking to block a vlote as long as they could keep standing. This is real "Mr Smith Goes to Washington" type stuff [1].
This Senate refused to change the rules on this so here we are. Two proposals have been made:
1. Eliminate the filibuster entirely. It's pretty much only ever used to block progress (eg by blocking civil rights legislation in the 1960s); or
2. Require a Senator to actually filibuster by going back to the old system of holding the floor and refusing to yield.
Both of course failed. Why? Because the corporate interests that own Manchin and Sinema don't want that because that might allow some form of progress, all under the guise of institutionalism. But this is a post-facto justification. I guarantee you the second the next Republican president holds office and the Republicans hold a majority in the Senate, the filibuster is gone.
The current rotating villains of the Democratic Party couldn't even bring themselves to bypass the filibuster to pass legislation to protect voting rights, which shouldn't even be controversial. But of course it is controversial because voting rights might endanger the red wave of voter suppression.
None of this has to be the case. Schumer won't go nuclear on this either because Schumer represents the interests of Wall Street.
[1]: https://www.youtube.com/watch?v=zPv0S1-ETdI