Here’s a plan.
Since Egypt has a long history with Yemen, even to training the Yemeni army long ago, why doesn’t the USA start a coalition (NATO/Japan) saying coalition countries will stop using the Suez Canal. Period. No threats. No bluster. Just quiet diplomacy. And we will brand the Houthis as pirates, not terrorists. One man’s terrorist is another man’s freedom fighter. But pirates are just pirates.
Moreover, the Coalition will refuse to accept cargo from any vessel that uses the Suez Canal. That makes things hot for China. If Xi is smart, he starts putting quiet pressure on Iran to stop arming the Houthis, because China will lose cash. Once Egypt starts losing $8 billion from canal fees it has incentive to exert force on the Houthis. Moreover, Egypt has a long history of military cooperation with Yemen, so they know who they are dealing with.
Make it a war on piracy, waged by an international coalition, ala the successful Kuwait War.
In addition to bombing Houthis strongholds, the Coalition will blockade Yemeni ports, cutting off supplies from Iran to the Houthis. Which also means starving Yemeni woman and children because Iran will try to smuggle arms in with the food, meaning everything must be blockaded.
So, use the historic tactic for Middle East wars: arm the Houthis’ Yemeni enemies. The Yemeni war is, to a degree, a tribal war. So, arm the other tribes. Suggest the Egyptians/Saudis give these “good Yemenis” cover to operate out of a defended area around Aden. Have the Egyptian Army/Saudis/good-Yemenis go in to do the dirty work of rooting out the pirates ashore. Boots on the ground. Remember, the Houthis would have broken Egyptian’s rice bowl, so the Egyptians will be motivated.
Bring UN humanitarian supplies into Aden to feed the “good Yemenis”. The Houthis will starve, and the Yemenis they control, faced with starvation, will tend to revolt against them, leaving the Houthis with little energy for attacking merchant ships. Those tribes that revolt get fed.
The Houthis will attack Aden to get to the supplies and there the Coalition/Egyptian/Saudis can help the “good Yemenis” to defeat the Houthis over a course of years.
The Egyptians/ Saudis, who will take will want something in exchange for cooperation, and for losing troops in ground assaults. So, the US administration will quietly tell the Israeli government to get out of Gaza. Enough is enough. The Israelis made their point. If they want to continue, the supplies of arms to them will become a trickle.
Blowback: In Western countries inflation will escalate as shipping costs rise. China may retaliate against the plan by raising tariffs on US farm imports. Also, the arms we give the “good Yemenis” will inevitably be used against us some years down the line, somehow, somewhere. But that’s the way things are handled in the Middle East, since before the time of the Persians.