The flat deck trailers are fine behind a F150. I pulled my 16' with a 4.2 5 speed 1997 F150 for a good while. And before that, a '95 F150 with a 4.9 big six. Both were 3.55 gear. The 4.9 pulled 10x better than the 4.2.
In 2006 I picked up a 24' Pace enclosed. Had to pick it up about 150 miles from home, and it's a little bit hilly up there. Not mountainous. Just hilly. Pulled it home with my dad's 2000 F150, 5.4 with 3.55's. 2wd. The motor loved it. The rest of the truck not so much. The tires SUCK with that much weight. P255/75R16, IIRC. P, to me, means passenger car tires. LT's are a must with that much weight. That day was windy. Trailer alone weighs 3500 lbs and is rated for 7000 max. I was driving west and the wind was coming out of the North at about 20-25 mph. It was all I could do to keep it in one lane. I said no way is this going to work once I get the car and all my junk in it. I stopped on the way back to grab some McDonalds grub and got on eBay with the phone, searching for an F250. Same city, a guy had a '95 F250 2wd, 7.3 auto long bed. Went to look at it, was pretty nice truck, 170,000 miles, ran fine (noisy), brand new Ford transmission at 168,000. Only problem...3.55 gears. I ended up buying it but couldn't pick it up til the next day. So I called a buddy who also lives nearby and had him store my "new" trailer overnight. Next day, dad and I went back to get the '95. Hooked it onto the enclosed and pulled it home. The difference is......well daylight and dark. THe brakes are 10x better. The chassis is built to hold more load. The engine doesn't struggle as much. Uses less fuel (but diesel costs more per gallon, unfortunately). Everything about the F250 is/was better than the F150.
The F150 was rated to pull I believe 8500 lbs. And it will. But an enclosed trailer that weighs 7000lb feels like it weighs 70,000 lbs if the wind is blowing much or an 18 wheeler passes you in the other lane. In the F150 it was scary. I pulled the enclosed (loaded down) with my '03 Lightning a couple times as well. Engine didn't work much and the 4R100 wasn't phased. The chassis, tires, and suspension were. I knew it was too much for it but had no choice since the '95 went to my brother. So I had to let the Lightning go, traded it on a '03 F250 7.3 (thank God it's not a 6.0) 4x4 crew Lariat. Love the truck. Pulls 10x better than the Lightning with that trailer. I actually feel safe pulling the big trailer with my F250.
Oh, and I would HIGHLY suggest a weight distributing hitch. Once again, comparing it to pulling directly off of the hitch to a WD hitch is like daylight & dark. I use an E3. Actually on my F250, the receiver is rated for 6000 max and 600 tongue. My trailer weighs 7000 and carries 690 lbs on the tongue, so I picked up the E3, took it to work and set it up in the parking lot after hours. On the way home, I immediately noticed that everything is super stable, no sway, weight on the front wheels instead of all on the rear wheels. Steers better and the headlights aren't in the clouds. Highly recommended. Expensive but worth EVERY penny.
And yes I still use my 16' deck trailer sometimes. I actually pulled the car to and from the track many moons ago with a '91 Ranger, 3.0/5 speed. That was kind of a wild ride. The F250 doesn't even realize it's behind it.