I can't think of a downside unless you crank the antisway bar so tight it forces the wheels to skid on the pavement. I can't count the number of tow-behind cement mixers I have seen being puled by trucks and they oscillate left and right about a foot every 10 seconds. They definitely need anti-sway bars!
Yes, and with respect to your analogy, keep your pants in place!
No a sway bar won't hurt anything. Other than add a little bit of weight. Nothing wrong with a belt and suspenders.
But in the case where the trailer tongue weight drops the rear of the tow vehicle too much it creates a dangerous situation by taking weight off of the front wheels which leads to reduced steering control reduced braking and sway. In this situation you need weight distribution. Sway control alone won't help.
The only situation where I've heard that weight distribution can't be used is on small SUVs that are unibody construction instead of a true frame. The WD puts more stress on the unibody that it can handle and may damage it. In that case you would need a different tow vehicle.