Of course, it's not for the US government to dictate what the UK should or should not do, no more than the UK should tell the US what to do, but the US DOES have the right to state its' own opinions and preferences, particularly if a decision by the UK affects US interests. If that influences the path the British take, then so be it. Personally I think these statements are as likely to hinder as much as persuade. After all, no one likes being told what to do by someone else. If the great British public thinks the US is trying to tell them what to do, they are as likely to demand we do the exact opposite!
In this case, it suits the US for the UK to be closely attached to Europe. It doesn't suit the UK, for a very great number of very good reasons, but that of course is not the concern of the US. The US government is predominantly interested in promoting American interests, not British ones, which is as it should be.
The reason why the Americans like the Brits involved with Europe is that they provide a useful link with the still very important European countries. Britain has a "special relationship" with the US (based on shared history, language and culture), but the rest of Europe generally does NOT have a special relationship with the US (different language, culture etc). Therefore the Brits can explain to the Europeans what America is about, and explain to the Europeans what the Americans are about. A Britain heavily involved with European decision-making is better able to provide that link. The British are far more likely to follow Washingtons lead on an issue than, say, the French.
On a personal level, I haven't heard any of this spiel being related in the British press. The great mass of the British public is probably unaware of this statement, so I don't think its had much of an effect. Of course the British foreign office will have understood what is being said.
I wonder why it had to be a public announcement? Couldn’t ithave been done some other way?