Your analysis is a little simplistic. Iran is Shiite Muslim and Persian ethnically. Most of the Arab states, as their name implies, are Arabs not Persians. In addition, they are mostly Sunni Muslims. The majority in Iraq are Shiite, though Arab and not Persian. Yemenis are 65% Arab and Sunni. They are roughly 35% Arab and Shiite which is the group from which the Houthi rebels are organized.
Historically there has been extreme hostility between Shiites and Sunnis as well as between Persians and Arabs. Because of the complicated tribal politics of the Middle East, strange alliances have developed over the years. Iran really has no love for the Houthi rebels, even though they are Shiite because they are Arab. But they have a total hostility towards Arab Sunnis, especially those supported and friendly with Saudi Arabia, who is a mortal enemy and the biggest counterpoint to Iranian hegemony in the region. Thus, Iran backs the Houthi rebels, not because of what they have in common, but because of their hatred for Saudi Arabians and desire to support their troublemaking along the shared border with Saudi Arabia.
A similar dynamic is at play in their support of both Hezbollah and Hamas. It is less motivated by a shared form of Shiite Islam than by a total antipathy towards Israel. Ethnically, the Arab Palestinians and Lebanese have no love of Persian Iran. What unites them is their denial of the right of Israel to exist and a program to utterly destroy the Israeli state. Ethnically, the Arab Palestinians and Lebanese have no love of Persian Iran.
It is not clear that an American warship shooting down Yemeni drones and cruise missiles supplied by Iran will have much, if any, impact upon the politics of the Middle East. The countries that have warmed up to Israel are the Sunni Arab states. The US intervening to prevent an attack by Shiites using Persian missles is likely to have little effect on their policy. They have already implicitly acknowledged the right of Israel to exist since they are engaged in negotiations with them to normalize relations.
The Arab states understand that it is Iran and its extremist proxies that are creating the instability in the Middle East. For peace and prosperity to become a reality, this is the real threat that needs to be controlled. The actions by the USS Carney have done nothing to change the calculus.
