AMRAAM was more than just natural development: natural development is something like sidewinder evolution.The AMRAAM was a natural development from the deficiencies associated with the Sparrow primarily by the lack of an active terminal homing.
Look at the size of their respective rocket motors.
Sparrow(and now, ironically, PL-15) was simply a much heavier missile, which could gain even more from improved chemistry and much lighter avionics.
But the point was to give 100% BVR capability to originally WVR-only F-16 family(by far the most numerous US and NATO fighter), instead of focusing on getting even more performance from F-15 family.
Navy, on the other hand, had no such plans: the plan was to squeeze out every inch of performance from the F-14D. +their lightweight fighter, the Hornet, was originally better at dealing with bulkier payloads than the Falcon and wasn't handicapped even by sparrow weights, so it could accept it as well.
But with the end of the CW, F-14D was dead, and wannabe "AIM-152" was very soon dead, too.