I was in Ipswich about 20 years ago when there was still the American military base RAF Woodbridge/Bentwaters. The American GI's were quite accustomed to the fact that many British girls were interested in them and the US in general. In fact, at the dances sponsored at the base, there was hardly any standing room from all the local girls who came. I had 2 British girlfriends who married American serviceman, and my closest British girlfriend became a nanny in the US to get a chance to live here. I don't think they were looking to 'get a green card', but I think that the British/American attraction goes back a long time (WW2 and beyond?) and it seems very natural whether it's Brit man/American woman or the opposite.
All that said, I think there are plenty of people who will marry to get a green card, but I also think that for many others, it may be a hindrance to a relationship. As many people can attest to here, a LDR and moving halfway around the world isn't that easy, so there has to be a strong foundation in the relationship to make it work.