Without an airborne radar platform, you are limited by the horizon of the earth. This means that you will not be able to detect low flying targets further that ~20 miles away (depends on stuff). So a low flying airplane with 20+ mile range missiles basically has a free ride to shoot at you. An airborne rader raises your radar horizon, which means that you can detect low flying targets (or ships) much farther away. This also lets the ships not use their radars, since the airborne platform handles that. Thus the
E-2s that American Carriers fly are probably the most important planes on the deck.
Next, missile defense is a very processor intensive process, you need to decide if you have been shot at, by what, how many missiles, what to fire at them, did you hit them, should you ifre again?... all in what little flight time the missile allows you.
The type 42 crusisers that the British fielded were (iirc) mainly created to shoot down planes, not missles. So they had real trouble shooting down the (modest) preforming Exocet missle.
So, no early warning radar, insuficient processing power, both for the missle interception plots and for the radar itself, and then so-so intercept missiles. Would have worked fine against planes trying to bomb the fleet, not so well against planes shooting standoff missiles.