Targeting key enabling (e.g. radar) or missile sites will absolutely have an effect.

The Houthis have large but limited stocks of missiles, little ability to domestically manufacture, and difficulty importing more advanced models from Iran at scale.

Moving interdiction left of launch is absolutely the right call, as it leverages weaknesses in Houthi air defense to more economically blunt their launches.

Hell, the US/UK could just close the loop on foreign military aid and buy low-cost Ukrainian UAVs to attack Houthi targets.