I had and 06 F 150 with the v6 and 4 speed auto. I added the tow package( oil cooler, battery charge and brake wiring etc).
I tow a 21 Amerilite. It weighs about 4500lbs loaded.
The truck didn't have any problems with it starting out or on the highway.
That said I don't think I would want to go much over this.
If the factory said it can tow 6500lbs I would guess the most you would actually be COMFORTABLE towing would be around 5000.
For me, being a safe and comfortable tow was important.
If you are on edge with white knuckles the whole trip, it won't be any fun, and you won't use it.
Remember the tow rating is based on a flat bed trailer loaded with cement blocks, not towing an 8' x 10' sail sideways.