Yup, Biden wants the fighting over before the campaign starts in earnest.
The problem is that despite what the West thinks, Netanyahu is right about a Palestinian state. As long as the West demands that Israel agree to some new political entity on its border that is going to hold elections, the truth is that sooner or later that entity will vote Hamas or another movement like Hamas back into power, and so the Israeli public, who also know this, will support Netanyahu in opposing that answer. As terrible as Bibi has been and is, after 2 decades he and Likud have succeeded in creating the exact reality on the other side that he needs/wants to justify his intransigence.
Not that I have any clue as to what an actual solution might look like either. 100 years ago the Western powers might have set up a protectorate and ruled it with a benign but iron hand to keep the radicals suppressed until they all died out, but no western nations have the stomach for that now - either to act with the harshness required or take the inevitable casualties if they didn't.