To understand this, look at where the Houthi's have been expanding: Taiz,
Marib* and Al-Jawf.
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Losing Taiz would be a blow for the Saudi backed Yemini government, however any capture of
Marib would be devastating for Saudi Arabia, if the Houthis were to push into Marib and hold that territory, they would have access to the Safer oil field, this is a nightmare scenario for both the Yemini Government and KSA.
The Saudis have been approaching neighboring nations including GCC, there is a widely held belief that the US may assist the Saudis in strikes to target missile launch and storage sites used by the Houthis particularly in the area North of Sana. Apart from this, there is a concern among many that airstrikes alone may not been enough.
The Saudis have learned that any cease-fire or negotiation window only gives the Houthis a window to re-arm and re-group before coming back stronger, the increasingly deep strikes into KSA have not only caused loss of lives, material and economic loss, these attacks whilst not particularly militarily effective have had the effect of embarrasing the Saudis as a stable partner to do business with.
The increasingly brazen attacks against Saudi Arabia have led to a change in tone among not only nations within the GCC but far-a-field as most now see this as a coercive attempt to throttle global economic output and fuel security.
With the EU currently embattled and entrenched in a war on its own borders and facing gas and fuel security, it is expected that some EU nations will likely increase their military support to the Kingdom in the form of ADS, Refueling and targeting systems. The
French, Greek already have patriot and ADS batteries operating in KSA with two further batteries brought into Saudi today:
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The UK are operating voyager aircraft which recently upgraded its boom[1] to supporting air-refueling for KSA F15s and Eurofighters operating in A2/AD operations and conducting strikes.
Regardless if the US enter into this conflict with aerial assets or not there is a widely held and popular opinion that air-strike along will not be enough to dismantle the Houthi capability to launch deep strikes into KSA or seriously threaten international shipping lanes across the Red Sea.
The Houthis have recently shown that the Burkan 2-H missile is capable of targeting deep inside Saudi sovereign territory:
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And their Samad-2/3 drones have been effective in targeting Saudi backed forces inside Yemen alongside targeting Saudi infrastructure inside KSA itself:
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Due to increasing number of missile and UAV threats posed by the Houthis and their ability to strike maritime and ground assets associated with Saudi Arabia's oil exports - the popular opinion among many analysts is that this may result in some form of ground intervention which is the only way the Saudis believe they can degrade the Houthi capability to target KSA vital infrastructure.
[1]
https://ukdefencejournal.org.uk/raf-aircraft-finally-to-get-boom-refuelling-capability/
