But Foxbats will actually stay there, while AWACS will come crashing in flames in combat
Technically, we don't have that guarantee on the Foxbat too
Range wise, the AWACS will never really be close to the forward line and it will always have some sort of HAVCAP protecting it.
If a Foxbat is engaged in the role of an AWACS, unless we put a rotating dish on it, it will only be able to look at objects with its forward line of sight, limited by the radar cone. A time will come where it will have to turn away because it gets too close, and during that time there would be a blackout unless another Foxbat is also engaged in providing targeting information.
Past this hurdle, there are multiple controllers sitting inside an AWACS controlling multiple lanes, and all of that responsibility is then shifted over to the pilot(s) present in the cockpit - he has to fly the plane, map out his safety, perform maneuevers, utilize the radar, process lots of requests, keep track of every flight in his area, and so on - a surefire receipe for helmet fire.
The AWACS crew can manage all of this efficiently and with speed, with only one plane airborne - and that's not counting its own defense arsenal: it can simply fry the fire control radar of a perceived threat by focusing all that power into one area, if something does get too close(assuming it's not taken care of by the HAVCAP already)
These are what BVR lanes may look like hypothetically, with each controller sitting in the AWACS managing one or two:
It's not the best sketch, but you can see how the single or dual Foxbat crew would be not as efficient in this environment as the AWACS.