One issue is most people buy pickups like the F150 cause they don't like driving little cars and so many trucks are now designed with engines,transmissions and rear ends for mpg. As mpg raises the load capacity/tow rating decreases. The new government mpg standards for fleets compounds this issue. The big 3 also figured out the their profit margin rides on the luxuries and upgraded interior packages. Why trucks come with auto and not manual transmissions standard. Why A/C and 4x4 trucks are so common. So if a "1/2 ton" pick averages around 8,200# GVWR (around 7,000# pre 2005) all the extras subtract from that, ei less payload. In the game of who can haul more for the PR Dept. (think Tundra towing space shuttle) the "1/2 ton" market has bleed over into the "3/4" market which was normally limited to 10,000# GVWR for non commercial registrations etc... So the "3/4 ton" in some ways was a castrated "1 ton" on paper for tax purposes.
I haul rear axles for GM and they stock 18 different axles for the 1500, 2500 and 3500 model trucks. Each axle will effect load and towing capacity.
You never stated what size trailer you were looking for. Find the trailer and shop the truck. A F150 and F250 are same body it just sits on a different frame, drive train and thus the F250 sits about 2" taller due to the stronger frame and drivetrain. If you get a F150 HD you will have the ride of a F250. Trust your math and don't rule out a F250 if price and need is there.
Also remember that boat trailers average a lower tongue weight (5-7%) than a travel trailer(12%) thus all the boats in the advertisements. So a tow rating of 9,000# might work for a boat but might only mean a 5,000# TT.