Non-Rhoticism always existed, even in America. In fact, the more historic the place in America- that is the east and the south, the stronger the non-rhoticism. New York, Boston, and New England, for example, are famous for their non-rhoticism. But, it should be noted that their non-rhoticism is of a different variety, that is, 'store' is pronounced more of 'sto-ah' with two syllables instead of one. Also, Shakespeare, while he did occasionally spelt 'colour' as 'color', sometimes spelt 'colour' in the fashion of the modern Brits. In fact, the French alternated between 'color' and 'colour' during this time as well. It was during the 18th century that any attempt was made to codify the English language in spelling and grammar. Indeed, Nathaniel Hawthorne spelt 'colour' with the '-our' as well. Webster is to thank for that development. Webster attempted to make English of the Americans more phonetic, 'color', 'center', &c. But, also note, he tried to force Americans to give up the '-ough' combination as well, spelling 'Cough' and 'Coff', 'Plough' as 'Plow', 'laugh' as 'laff', &c. Many of his attempts at reformation were not very successful, and only recently making a revival on the internet by people who can't be bothered to add two more letters to make 'you'.
Additionally, most Modern British Accents can be said to be more 'closer' to the English of olden times, places like Yorkshire using 'thou' and 'thee' still, and sometime mixing in the Norse of the Danelaw. Rural Britain usually do not see many people coming in, therefore, their accents remain unchanged or completely diverge on another path altogether, hence the famous anecdote of Yorkshire men, ie that from a man from one village haven't a clue as to what another man from a village two miles away is saying.
American English, on the other, seen a different sort of development- the adoption of Indian words, the influx of Immigrants, the African-American, &c. If I were to speak to someone of the 17th century, say John Bunyan of the Pilgrim's Progress, he probably would understand but two or three words from me, and I would understand, probably, an equal amount from him.
The main factor for this belief, I think, is the difference between our 'official' accents:
the official accent of Britain, the RP accent, was considered by the BBC because of its association with your 'Public Schools' (Private Schools in America) and the grandeur of the elite, whereas the Midland accent, that is the accent of our Newscaster, was chosen for its neutrality and for its worldly, earthly quality. Because many people associate the 'earthly' for the 'ancient', I suppose that is what cause this belief.