Does the president have a “duty” to put forth a nominee? Does the Senate have a “duty” to consider him or her? No.
I’ve been meaning to write about this but, as noted there, “shall” does not necessary mean that it is a mandatory act, and there is no time limit on it. The president has the power to nominate, and the Senate has the power to advise and consent, but it is not incumbent upon either to do so. The Framers intent was not to insure that appointees were appointed, but that it not be done unilaterally by a single branch. In their desire to limit government powers, they made appointments, as with much else, difficult, and I doubt they’d be displeased by either the “Biden rule” or the current Republican stance.
The only real duties that either branch has are to “establish Justice, insure domestic Tranquility, provide for the common defence [sic], promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity.” It is ultimately up to the people to decide if they are doing so, and to rectify the situation if not.
If the president chooses not to nominate, he will be judged in the next election. If the Senate advises and consents by advising the president that they will not consent, they too will be judged in the next election, or at least a third of them will be (as it happens, the Republicans have 24 seats at risk in the fall). But the Senate is doing nothing unconstitutional in deciding to let the people decide.
[Update a while later]
Welp, Kasich said today that he might nominate Garland. So we have that going for us. #RINO