Can you point to the law McConnell violated in doing this? And if not, in what sense is the current court illegitimate? Finally, how does this in any way address my question about the president "seizing" power?
It's the Senate's Constitutional responsibility to advise and consent the President's SCOTUS nominations. What is the procedure when they abdicate that responsibility? Remember, the Senate didn't tell Obama "no, we don't like your nominee." They, meaning McConnell, refused to hold any hearings on at all.
So maybe he didn't violate the law, but he certainly didn't uphold his Constitutional responsibilities, which I believe is part of the Oath of Office he took.
That is a distinction without a difference in this context, which is about the Senate abdicating their constitutional responsibilities. They didn't hold public hearings before 1916, that is true (the history of why they even started holding public hearings is pretty controversial in itself.) However, they still held yay/nay votes on the floor for the nominees, in order to fulfill their advise and consent role. McConnell refused to hold a vote, therefore he failed in upholding the Senate's constitutional role on the SCOTUS confirmation process.
That's not how the Senate works. They are not required to hold a vote in order to declare "nay" on a nominee. Declining to vote is effectively a "nay". It's no different than declining to hold a vote on a bill which they know will not pass.
This is how Congress and the co-equal branches of government are designed to work. Congress and the President often don't get along. It's a feature, not a bug.