The Middle East conflicts between Iran’s proxies and Israel are difficult to end. The two sides have understandable aims and also aims that are incompatible with peace. Neither is yet willing to compromise sufficiently to secure a negotiated settlement.
Hamas and Hezbollah are hybrid parties. Both have extremist terrorist wings. The extremists want to drive the Jews out of Israel. They are happy to murder civilians, take hostages, keep military men and weapons in hospitals or schools and embed armies in civilian areas to make it difficult for Israel to counter them
There are more moderate Palestinians who want an end to civilian deaths in the war, an end to Israeli settlements spreading, the complete withdrawal of Israeli troops and the creation of a Palestinian state.
There are Israelis who think there can be no peace for Israel until all the Hamas and Hezbollah terrorists have been killed, all the weapons destroyed and Israel plays a part in policing Gaza and parts of Lebanon to prevent their reintroduction. They favour continuing the use of great force until the terrorists have been eliminated. If this also involves pre emptive action against Iranian nuclear installations and missile capabilities, so be it.
There are more moderate Israelis who want their hostages home, and some negotiated peace that gives them some guarantee of no more rocket attacks.
In order to pull off a ceasefire followed by a peace the Middle East and US interlocutors need to get the moderates on both sides to hammer out a common approach and to marginalise the more extreme elements. This is not proving easy. it also leaves open what the Iran facing terrorists might do to upend a peace.
The US and Iran both have considerable leverage. Iran could instruct her proxies to stop sending missiles and terror groups to Israel but does not want to. The US can influence Israeli military activities by offering help as with the shooting down of incoming missile attacks when the two are aligned, or witholding weapons and support if Israel goes too far in her aggressive responses.
US foreign policy was altered substantially by Joe Biden when he became President. He dropped the Trump strategy of creating a strong alliance between Israel, the US, Qatar, UAE and Saudi Arabia against Iran. He relaxed the attempt to starve Iran of money by sanctions to make it more difficult for her to finance an arsenal against the democracies, to train and arm proxies and to develop a nuclear weapon. President Biden decided to join the EU in trying to get a negotiated settlement with Iran, but failed. More recently Joe Biden has been trying to reconstruct the US/Arab/Israeli alliance against a background of Arab states hostile to Israeli military activities against Palestinians.