In one sense, you are right. Filibusters are not mentioned in the Constitution. However, the Constitution does give the House and Senate the right to make their own rules. Article I, Section 5, clause 2 says, "Each House may determine the Rules of its Proceedings..."
Filibusters arose very early in the nation's history as part of the rules of each house of Congress. The House of Reps basically did away with it in their rules long ago, but the Senate has kept the filibuster in one form or another since the 1830's.
I don't mind the filibuster in principle, but loathe the way it has evolved since what I would call the phony filibuster came into being in the second half of the 20th Century. It allows a senator to just declare one for almost any reason. The senator does not have to stand and speak on the topic continuously until s/he gets too tired to continue or gets his/her way. Filibusters originally were only intended to prevent the passage of bills, but in the phony filibuster era, senators could declare one for almost any reason.
Now we have the absurd situation where almost everything in the Senate requires 60 votes to pass. So yes, let's get rid of phony filibusters. If the Senate insists on keeping filibusters, then return the practice to its original form. Require the filibustering senator to do it in person on the Senate floor. Also amend the rules so that the filibuster is returned to its original more narrow purpose of only preventing passage of a bill.
Sèe #34
Filibusters originally were only intended to prevent the passage of bills, but in the phony filibuster era, senators could declare one for almost any reason.