Iirc AIM9 had a very good hit rate In that war but even so only 2 weapons per aircraft means that a pair can only engage a max of 4 targets likely a single wave or part of a single wave with limited options to take follow on shots of the first fails and then return to the carrier.
The whole purpose of the CAP was not primarily to knock Argentine planes out of the sky. It was to disrupt Argentine attacks so that the unloading could proceed more or less unhindered. Knocking down Argentine planes was a bonus easing the task next time. But the principle objective was to get the Green and Red berets and their equipment ashore. Once that had been achieved, it's game over, and all that remains are the details.