I ride on quiet roads with my kids in the trailer, but tend to avoid anything too large and busy. Drivers tend to react much better to cyclists with trailers than lone cyclists.
The main concern I would have with using a nice road bike to pull a trailer is the gearing. IMO you absolutely need a triple to pull a trailer with children in it up any sort of hill. My lowest gear is 28-28 and it is pretty borderline on some hills when 2 children + accessories are loaded.