I find that hard to believe, especially since a large part of the immigrants from Europe weren't even English speakers anyway.
New York was founded by the Dutch for example. There are a lot of Italian, Polish, French and Chinese immigrants in the US... English descendants were not a large majority.
I think it's far more likely that the US dialect of English is a result of foreigners trying to adopt the English language. Compare for example the situation in India or Jamaica. They speak English there, but with their own distinct accent on top.
Other than that, not everyone in Britain speaks 'posh'. There are tons of old dialects. I think especially the Irish had a big influence on how US English came out.
It also makes more sense in the light that other English-speaking parts of the world, such as parts of Africa, Australia and New Zealand, are closer to British English than the US dialect.