You realize that the "Yemenis" are an Iranian client state? Anything major from there is on orders from Tehran.
So people keep asserting.
But the reality of revolutionaries, insurgents, terrorists, freedom fighters, whatever you call them, throughout history, is that they fight first and foremost for themselves, their communities, families, and beliefs. State sponsors and supporters often like to imagine that they are pulling the strings, but the fighters will tolerate being told what to do only so far as it matches their own objectives; And the relationship is invariably more about supplies and logistical support than it is about command and control.
In detail you are right. In general, though, if you don't do what the guys arming you want you aren't going to get the weapons.
However, this is a special case anyway--not your typical guns, but high tech weapons. The Houthis pretend they're building them themselves but that's a joke. The drones have Chinese engines and everything else is Iranian. These aren't the toys you can buy off the shelf--such toys don't have the lifting capacity to haul an appreciable warhead.