Where do I begin...
A. F150's are not 3/4 ton trucks.
B. GCWR and GVWR are not the same thing. GVWR (Gross Vehicle Weight Rating) is the max weight on the trucks 2 axles, GCWR (Gross Combined Weight Rating) is the max weight on the truck and trailer.
C. GCWR has no affect on in the bed payload capacity, this is solely based on GVWR. Payload capacity is GVWR minus the empty weight of the vehicle.
D. Gear ratio doesn't affect GVWR.
E. GVWR on an F150 ranges from 7600lbs to 7850lbs
Honestly your dealer should have known better as they lost a sale. If you aren't intending to pull a trailer with a service truck any gear ratio will be fine.
I agree with everything you wrote. Except “D”. Gear ratio ABSOLUTELY affects GVWR.