Let the jihadis have the KSA. At some point they are going to have to govern and make policy decisions. With all the westerners (and assorted other infidels) gone they will not be able to keep the oil flowing and the local economy will fall apart. The gulf emirates will move even closer to the U.S. and some of the governments in the region who are not now friendly towards us (Egypt for instance) will likely move closer to us as they are unlikely to want their citizens getting any crazy ideas about revolution when they go on a haj. Right now the Saudi princelings are not our friends; I find it hard to imagine that we will be worse off with avowed enemies in charge as opposed to enemies who pretend to be our friends.
I think that could be a bigger problem then you allow. It would disrupt the world oil market, and that has real consequences. But with control of the KSA comes a military. Not competent by western standards, but if Jihadis control the SA military you can bet that the UAE etc will fall militarily immediately. Kuwait could be protected, but the rest of the sheikhdoms will be invaded.