The dealer is wrong. The police can not pull you over just to see if you have working brakes or a battery on your trailer. Even if they pull you over for speeding, they have no reason to ask if your brakes are working on the trailer.
However if the trailer breaks away from the car, and does not come to a stop on it's own, then they will begin asking that question. A 2,000 pound utility trailer would not cause much damage without the brakes not working. It is a 5,000+ pound travel trailer that can go for hundreds of feet without brakes, damaging anything in it's path.
I think the mechanic was trying to upsell you something, and causing excess worry. Sure it is a 'good idea' to have brakes on your trailer, and it will help you stop faster. But legal requirement? No. Chances that you will ever get asked about it randomly by a police officer? None. Chances about the Police inspecting your car and braking system if you are in a accident? Pretty good.
Good luck!
Fred.