Will Baude has an interesting post on various aspects of what the Senate Democrats did, which is well worth reading. One question he asks is whether the Senate Democrats passed a new rule or simply refused to apply the existing rule, perhaps creating a precedent. Another question involves the power of the majority of the Senate to make decisions, without being bound by existing Senate rules.
The constitutional problem is that the current Senate rules purport to stop a majority from changing the rules, by requiring a 2/3 majority to get cloture on a proposal to change the rules. It is the combination of that 2/3 requirement and the 60-vote cloture requirement that is unconstitutional. And it is not clear to me, in the case of such an unconstitutional combination, which rule is supposed to give way. One possibility is that the constitutional remedy is to jettison the 2/3 rule-amending requirement, and then use that power to formally amend the filibuster (if desired), not just to ignore the filibuster rule.
This is an interesting question, and I think there are arguments on both sides of whether the Senate majority has to change the 60 vote cloture rule or whether it can simply ignore the existing rule.
Ed Whelan also has three interesting posts on the filibuster, written from the perspective of a Republican who believes the Senate Democrats have been very hypocritical. Whelan notes that he has long favored eliminating the filibuster on judicial nominees, which is true. I first met Whelan when he I debated whether the appointment of Supreme Court justices should be subject to a supermajority rule.