What I meant by "broke the filibuster" is describe in this paragraph from the linked article:

Under the old rules, cloture could be voted by two-thirds of members present and voting. Fifty senators were required for a quorum. Do the math. If the only member of the filibustering party to show up for the vote is the senator holding the floor, fifty senators from the other side could show up and end the filibuster, as they would represent 50/51 of the senators present and voting. If only twenty senators show up to support the filibuster, forty-one senators could vote cloture. Under the new and current rule, it takes 3/5 of sitting senators to end debate. That’s sixty senators, no matter how many showed up from the other side. As a result, mounting a filibuster requires no work at all. No one has to stick around to vote. No sleep is lost, no fund-raising time is lost, nothing is lost. That’s a big deal.

The filibuster "worked" because it was politically and personally difficult to mount, which meant it was used sparingly. It was "broken" by the rule change that made if so easy to use that it became automatic. That's the point of the charts I linked.

As for why there is a filibuster, it provides some stability as small majorities come and go, and, even when it is difficult to use, it prevents the majority from running roughshod over the minority on matters important enough to be worth filibustering. That's why I want to mend it, not end it.