That's quite a video. Apparently the guy didn't know to apply the trailer brakes to stop the sway. Apparently didn't know to tie down things on the rear either. Hard to say but was also likely towing too fast. I don't think it's an issue of a WDH or not.
Does the OP need WDH? No. He's under the the ratings of his hitch and truck without it. With the trailer loaded the tongue weight may be over 500 pounds. The truck's hitch may require a WDH above 500 pounds of tongue weight. If it were mine I'd just upgrade to a hitch that can handle the tongue weight and forget about the heavy shin knocker of a WDH.
If sway is an issue then you need to increase tongue weight or possibly use sway control. In the last 14 years of towing flatbed, boat, cargo, and Rv trailers I've used WDH exactly once. That was when I borrowed a guys camper and he already had the set up laying there.