I think in this case it probably wouldn't have made a difference if the controller had realised they had an emergency sooner, but that doesn't mean the discussion isn't valid.
From the outside looking in, it seems the problem with "declaring an emergency" is that phrase isn't "magic" enough. It can become lost in "ers" and "ums" and other bla bla. The words "Mayday! Mayday! Mayday!", followed by your callsign three times, is very difficult to misinterpret. And so what if the response is, "Are you declaring an emergency?" You can be sure the controllers ears pricked up immediately. If the Americans want to be different, well that's ok I guess, they're not alone in that, but at least pick a phrase that is clear and concise and doesn't get lost in a ramble of other words. "Beetlejuice! Beetlejuice! Beetlejuice!" would be better than "declaring an emergency."