Sea Harrier radar and the Falklands early-warning gap

Would CATOBAR Phantoms have made the British carrier force more effective in the Falklands?

Yes, but the Sea Harrier was already radar-equipped, and the more serious British gap was the absence of airborne early warning.

The Sea Harrier possessed radar and was not confined to visual-range combat as the question assumes. One squadron initially used its radar more effectively than the other, but both improved during the Falklands campaign. The later Blue Vixen radar made the aircraft a capable beyond-visual-range fighter, while the earlier Blue Fox also supported successful interceptions. A CATOBAR force of Phantoms and support aircraft would have offered greater capability. Britain had left the CATOBAR carrier business because it was judged too expensive, then retained a capable carrier force through STOVL aviation. Even so, the absence of airborne early warning, including a possible Sea King solution, is identified as a particularly important source of British losses in 1982.