Obama and his administration might think this way, but I really don't think the majority of Americans would agree with him. You should have seen the opposition to the North American monetary union... no one here wanted the Amero.
I've heard that the main reason the US wants Britain in the EU is so that we can indirectly influence the EU through Britain's vote. But I think if Britain actually left, we'd probably be a lot more wary of the EU in general. I wouldn't say the French or the Germans are exactly close allies. It's possible they're bluffing because they fear losing control over the EU. For instance, they might not allow us to continue marketing stuff with both metric and Imperial labelling if it weren't for the British influencing the situation. Also, there is a general socialist tendency within the EU, and Britain helps pull the EU further right economically, even if it means Britain gets pulled left.
I mean, think about this... if we said we supported a British exit, how would that affect our relations with the EU? I can't imagine they'd be happy about it. Even saying that we guaranteed Britain something if they left might step on their toes. And then what if we supported it, and Britain voted to stay in? What would we look like then? It's just not something that can be said without political consequences. But if we say something like what was said, and then Britain leaves, no one will blame us for it, and we can still pursue trading relations without anyone questioning it. You don't really know what will happen until the die is cast, so to speak. Saying you will or won't do something is a long way from following through when it actually happens.
Besides, I wouldn't say America is Britain's best friend... we're not even a Commonwealth member. Wouldn't Australia and Canada be closer?
I really think in the end, though, trade with countries that can communicate in English might be valuable enough to be a leverage point.