This isn't about Israel either. Do you seriously think the US would be so supportive if this alliance was about Israel?
I have a theory about this that I cannot prove.
I think the current geopolitical environment allows us to build a coalition that appears to be against Iran. Presenting it this way could prevent possible American opposition and might even help us gain support.
Under this pretext, Türkiye could expand its military presence and influence in Saudi Arabia and Yemen -effectively around the Bab el-Mandeb- by using the Houthi threat as a justification, while using Hezbollah as a reason to expand its presence in Syria and Lebanon.
In such an environment, Israel would potentially be the only country with which Türkiye would eventually come into direct confrontation. Until then, Türkiye could continue to present its actions as part of the broader effort to fight Iran and contain its regional influence.
I suspect there may actually be some mutual understanding between Türkiye and Iran on this issue, perhaps even some kind of agreement. I do not think the ultimate goal is to completely eliminate either the Houthis or Hezbollah. My guess is that both groups will be weakened to some extent, after which they will reach an agreement. And perhaps, one day, we could even find ourselves fighting on the same side.