You need an F350 for that trailer. Don't kid yourself. And it should be a dually, although you might make a SRW work.

The towing guides don't help you much with the most important rating, and that is GVWR. For that you need to look at the truck specifications, and the trailer specifications. An F250 tops out at 10k# GVWR, and a crewcab truck will weigh over 8k# by the time you load it with people, fuel, gear, and a fiver hitch. That only leaves 2k# for the trailer pin (less than that likely). The trailer you mentioned lists a pin weight of 2,515#, but that is for the empty trailer without options, and without your stuff loaded. Your pin will probably weigh more than 3,000# ready to roll.

An F350 SRW has up to 1,500# more cargo capacity. You have some math to do to know if that is enough. An F250 definitely is not.

Yes, there are a hell of a lot of overloaded trailers on the road.