If you declare a "fuel emergency", or "we have an emergency" in the UK, without explaining the details you will be asked to confirm as ATC in the UK would much prefer the formal declaration via the standard phraseology.
Yes, I am aware we are talking about the US here, and that "declaring an emergency" is acceptable phraseology in the US, however many US pilot fly in both arenas and being in the habit of using "USA only" radio procedures could result in confusion in other parts of the world.
Having said that, I am firmly on this Captain's side. The situation deteriorated to the point where it was necessary for him to take control of the situation to provide a safe outcome for his passengers - and a safe outcome was achieved.
Job Done.
As an aside, I have never seen a preferred runway system which stays in place at more than 15 knots crosswind, when a more into wind runway is available (and the usual standard is 10 knots). I am rather shocked that such a situation has been allowed to occur.