No. The brake controller has the brains to properly apply the brakes on the trailer in response to the tow vehicle. If you apply the brakes gently on the tow vehicle, the trailer brakes will also be gentle. A hard stop will cause the trailer to brake hard as well. The controller also allows you to adjust for different trailer weights. A heavier trailer will require more braking force than a light trailer.
I don't disagree with this, but I wanted to add a bit for the sake of clarity.
You're describing the best brake controllers out there, but not all have the ability to vary brake force like this. The cheaper ones are known as "time delay" style controllers, and they have no way of knowing how much braking power the driver needs. You set them to a certain point, and they increase brake voltage up to that point over a short period once the brake lights come on. Whether you're lightly braking or in a complete panic stop, the trailer brakes behave the same. I *HATE* these kind of controllers, but they're out there.
A good brake controller, like the venerable Tekonsha Prodigy line, uses an inertia sensor to detect brake force, and varies the brake voltage to the trailer accordingly. They're far from perfect, but the best solution for a vehicle that doesn't have a controller integrated into the braking system.
Electric trailer bakes will not function without a brake controller. You'll be towing a trailer with no brakes at all, as there will be nothing to supply voltage to the brake signal line to actuate the magnetic actuators.