I think an important aspect to remember is that the RN were very aware of the fact of how obsolescent the Sea Vixen was by the time it entered service.
They wanted the F4 with its advanced radar, Sparrows etc. and did not want to get stuck with another "old" sub-optimal interim aircraft.
And the blunt truth is that the F4 was significantly more advanced than the F8 and was also wanted by and bought for the RAF (who had no interest in the F8).
Apart from being the superior fleet defender the F4 was also so much easier to land on a carrier.
The decision to go for the F4 was also made in the expectation of operating from the follow-on UK aircraft carriers that were subsequently cancelled.
In the actual context of when the decision was made the F4 was a more logical choice.
While I appreciate the F8 as a fine aircraft and how one can be an enthusiast for a particular aircraft I think emotion may be clouding the arguments of some contributors.
So the first point here is if you want to discuss history, you're posting in the wrong section.
Second I agree that the F4 made enormous sense after the '63 Airdisplay by the Soviets showing their new large anti-ship missiles.
Hence not investing in what were thought more modern and superior aircraft to either F8 and the sainted F4.
Time had become critical and the F4 had enough future potential to put back the effort for wonder weapons.
Thirdly
It's actually the other way around as the FAA pushed for the F4 while the RAF stuck with the P1154 'Harrier' until '65.
Even then they afterwards tasked only 72 for MRI until the cheaper to operate Jaguar came into service.
FAA got stuck with the P1154 for the sake of commonality in imitation of the then US TFX concept.
It's not what they originally wanted which was OR.346, think TSR.2 navalised. .....
Fourthly
The context of any decision for the F8 is either:-
A less ambitious CV force. Such as focused on ASW with more limited fighter and offenaive systems.
Or that the F8 might be quicker to service.