I'm an old trailer guy in addition to an old car guy. I'm not saying there weren't hydraulic brakes out there that tied into the vehicle master (sounds like a nightmare to bleed), but I've never seen one. The closest I've encountered would be surge brakes, which have their own master cylinder on the tongue and are completely separate from the vehicle's hydraulic braking system. On the other hand, electric trailer brakes were very, very common back in the 1950s.
Assuming the vehicle and cooling system are in good shape and you're not crossing the Rockies or towing a 40' monster, you'll be fine. Trailer brakes are less for stopping power and more for trailer control- anything much over 1800-2000lbs and you'd be wise to use them.
Trailers prior to the mid-1960s were much, much lighter than modern trailers, thanks to the advent of cheap particle board (and the fact that folks tow with trucks instead of cars since the '80s or so), so while a 1957 16' trailer might barely crack 2000lbs a modern trailer of similar size would be well over 3000lbs.
I'd try to keep the trailer weight around 2000-2500lbs tops behind your Fairlane, and be certain your tongue weight is set up properly (that means no bike racks hanging off the back of your trailer, ever, shudder). A load-balancing setup would be a good idea as well.
Here's my '57 Cardinal, which has been neglected recently in favor of the '55 Courier that will be towing it.
