I know i am sound too much like a lawyer, but the right to waive is not the same thing as having the right to have a bench trial. One side can waive a right but they other side can assert it. but i hope i am wrong.
It is the defendant's right, not the state's, to have a jury trial.
If an accused has the right to waive a jury trial, he has that right. I don't see why there should be any issue. It is the accused, not the state, which has a right to a jury trial.
I trust the source, Martindale-Hubbell, and it clearly states the defendant can waive and opt for the bench trial. And that sounds right.