Hi belyaevg - if your brakes are applied by the tow vehicle or by pulling the breakaway pin, you should NOT be able to turn a trailer wheel by hand, at all. The brake controller in your tow vehicle should be able to lock the brakes in your trailer too, if you set the controller to maximum. The people at your U-Haul don't know what they're talking about.
Trailer brakes are NOT just "emergency brakes". They should act with your tow vehicle brakes to minimize stopping distance when you are towing a trailer. Essentially, you want them to exert just enough stopping force that you don't feel like the trailer is "pushing" you when slowing and not "pulling" back on you either.
It sounds like you don't have a complete electrical circuit from the tow vehicle to to brakes and back to the tow vehicle. Since you don't have another tow vehicle to experiment with, you'll have to do more complicated troubleshooting. Use a multimeter and probe the brake socket and ground on the tow vehicle plug while someone else applies the brakes, possibly with the ignition on, but the engine not running. You should see voltage that increases as the brake controller intensity is increased. If you don't see that, your problem is in the truck's wiring.
Set the multimeter to circuit continuity, and probe the trailer's plug at the brake and ground blades. You should see a complete circuit. If not, the trailer wiring is faulty. It could be a bad ground or it could be one or more of the wire nuts near the trailer wheels, which may have fallen off and allowed the wires there to be disconnected.
Let us know what you find.
- Jack