Im on mobile on the train so sorry this will be short.

The notion of one accent or dialect being considered high class is known as prestige. In america, the dialect of an upper middle class white person is a prestige dialect. The term prestige means exactly what you'd think, but is also a specific term for this concept in linguistics. A posh British dialect is also a prestige variety due to the cultural conditioning where the wealthy might have put on this accent even if it wasn't really how they would have spoken otherwise.

However, you're also mistaken about the accents of the 1770s. British people did not sound the way they do today, and neither did Americans. Accents on both sides of the Atlantic have changed considerably since then.

edit: typos