What is the three-light indicator on carrier-borne airplanes?
Those are angle of attack indicator lights. The landing signal officer on the carrier can judge whether a pilot making an approach of the boat is on angle of attack using those. The center light will be amber and will indicate that the airplane is on angle of attack for the approach. The upper light is green and will indicate to the pilot is it too high of an angle of attack, that is going to slow. The lower light will be red and indicates that the pilot is it too low of an angle of attack i.e. is too shallow in approach angle and going to fast.  The lights correspond with those on the angle of attack indicator which is located to the left of the head up display in the cockpit.

Flying the proper approach speed is a key ingredient of any approach be it carrier or terrestrial.  Again due to the inherent tighter tolerances of a carrier landing, the AoA data is useful for the LSOs to spot deviations in the approach.  Second, since Vref varies for all aircraft approaching the ship based on type, gross weight, etc. a radar tack of the aircraft would require the LSOs compute the required Vref ahead of hand for every pass at the boat.  It’s much easier just to observe a simple light indicator on the jet from the LSO platform to gauge this by.  

Regardless LSOs do also have access to the raw AoA information from an aircraft making a pass and it is displayed on the PLAT camera at the LSO’s platform.  Whether that data is based on a direct telemetry from the jet on approach or some other source is unknown.  One possible reason for retaining the lights is for Emission Control (EMCON). Radio signals known to be used for recovery procedures would be valuable to an enemy since a carrier is most vulnerable to attack during startup and recovery operations and a simple set of lights visible only in the immediate vicinity of the ship would be advantageous here.