The proxies in proxy wars always have their own motivations and agency (see also Egypt, Israel, and Syria in the Cold War; or North/South Vietnam). Doesn't change the great-power dynamics of both sides' sponsors.
I was thinking more of the Iran-KSA relationship; in the Middle Eastern context, the defining regional-power conflict is between Iran and the KSA. The anti-Houthi militias are more clearly Saudi and Emirati proxies than American ones.
That's complete nonsense. Neither the Hadi (Saudi-backed) nor STC (UAE-backed) types are particularly Islamist; ISIS and al-Qaeda are a third bloc in the rural/desert east that the US periodically bombs, while the Saudis are busy generally bombing Houthi-governed civilians.
That's the official Saudi line and it doesn't take a genius to see its BS:
This whole mess escalated when houthi rebels tried to stop ISIS from taking over the government. Where did those come from? What did the Saudis do? At what point did they started bombing civilians??
ISIS and al-Qaeda were not involved in any of the maneuvering around Sana'a, and the Saudi intervention kicked off when they realized that a) Hadi still had a power base in the south, and b) the Houthis were about to capture it
>ISIS and al-Qaeda were not involved in any of the maneuvering around Sana'a,
While true, in many countries a group like Al-Islah would probably be described in the Western media as "linked to ISIS" or something similar. They're part of the Muslim Brotherhood also regularly maligned in the West.
>The anti-Houthi militias are more clearly Saudi and Emirati proxies than American ones
They're still being supported by the US military, with billions in arm sales, tactical support, and a heavy involvement in the ongoing blockade. While the Houthi forces can claim rather minor support from Iran.