I think, first, that you should pay less attention to the detail of a TV documentary and read an authoritative report on the incident.
It was a whole lot more complex a situation than what you are relating in your post. Certainly there were mistakes made on the ship and lessons learned .. but it was not a cool, calm and laid back set of circumstances and there was mucho stress in the air ... a very good example of the man-machine interface and HF problems associated with high stress and high workload in the midst of carry over thoughts from a then-recent incident where a sister vessel had been significantly damaged by a inadequately defended air to surface attack.
Add to this the aircrew's presumably being a tad complacent and possibly not looking after their own SA as best they might have done.
Keep in mind that a TV doco brief does not necessarily seek to present a balanced view of the formal investigation but may, in fact, put an editorial spin on the whole thing.