Why do thrust reversers on jet planes work? By Newton's third law, it seems like the force exerted by pushing air through the nozzle should cancel the force from redirecting the air forwards. The problem seems identical to the case of a fan on a sailboat, with "fan" replaced by "jet engine" and "sail" replaced by "thrust reverser."
Thrust reversers are designed to reflect the air that hits them. The air does not simply leave the engine and stop when it hits the reverser. Instead the air is reflected and travels forward past the engine. The end result is equivalent to turning the engine round so it blows air forwards rather than backwards.
To use your example of the fan and boat: in this example you normally assume the air starts stationary, is accelerated by the fan, stopped by the sail and ends up stationary again. That's why there is no net force on the boat. However if the sail reflected the air from the fan instead of stopping it, the air would start stationary and end up moving backwards and there would be a net force on the boat.