Counter-battery fire is tremendously precedented and always done at standoff. You can also start hitting arms stores, port infrastructure, training and C3 facilities.
The beauty of this is it’s cruelly win-win-win. The Houthis can use the bombing to strengthen their domestic image, maybe even boost recruitment. Iran can piggyback on that. And America can claim it cleared the Strait. As long as everyone stays in their lane (literally), it’s a stable conclusion.
Training and top level command is most likely not even in Yemen. You could hit the ports, that wouldn't stop the import of these missiles - they are shipped in small boats as a kit, assembled wherever, and then kept in a cave somewhere or in a car, ready to be fired. No port or infrastructure needed.
These tactics have been used since the 80s, and no solution short of a ground invasion can stop them. Israel couldn't even stop Hamas and the PIJ from firing guided rockets from Gaza - at the end of the day when the IDF bomb houses that rockets were fired from, it's just theater: they never store more than a dozen munitions, and by the time counter battery fire arrives, they've likely fired all munitions already. This is a tiny 2.4sqm strip fully blockaded, I don't see how you can stop it in Yemen.
This is blind counter-battery. You use the shot to place loiters. That then trains your fire.
Despite much media ado about destroying the launchers, those are typically just welded pipe or wooden catapults. There's no value in them and no point in ever coming back there.
Those arms stores do not exist as such, they're most probably highly dispersed and only at the limit can one call them "stores", and if gathered in one place that place is most probably located underground, where aerial bombings would have close to no effects.
Just look at how difficult it is right now for Israel to take out Hamas's weapons cashes in Gaza, and we're talking about a much concentrated operation in terms of space and most probably Israel knows a lot more about Hamas's weapons caches than the US would be able to ever know about where the Houthis store their weapons.
See the thing is, if shipping companies don't trust the waters they simply won't send ships there - claims don't matter one whit.
What part of removing long-range precision strike capability suggests an empty claim?
Getting information on where things are stored is hard. It needs boots/spies on the ground (satellites can only get limited information and are easily fooled). As pointed out elsewhere, modern best practice is to not have a large warehouse that is easy to find and destroy, instead you scatter this stuff around in small units.
If they launched their entire stockpile, it’s no longer an issue. The question was using one launch to take out a couple missiles, maybe a launcher and those operating it. Done repeatedly, this will degrade a static force. (Additional levers would need to be pulled on resupply.)
When it comes to missile (whether drone, cruise missiles, or ballistic missiles), it's much more difficult. The US had a hard time countering the SCUD missiles in 1991. And that was with a huge military, with the best ISR the world has ever seen. A 6-pack of Shaheds can be launched from a trailer that looks all the world like a normal flatbed style semi. A cruise missile can be fired and navigate a course that obfuscates its launch point. Ballistic missiles can fire, then drive away to hide in a city.
The only way to win this fight is to blockade the country from receiving shipments from Iran. Air blockade and naval blockade, combined with strikes to hit known depots etc. This is called war. Not a presence mission, not a "response," but the literal definition of war. The US has no stomach for this entanglement, and hopes that the problem goes away. It won't, but the pain level is relatively low, and the USN is getting some great practice in fighting a LIC in the littoral regions.
So if hitting the launchers is hard, how about destroying one of those assets every time they attack shipping?