Pay-shio? People really say that?!! Eeewww...
I'm not quite sure what all y'all mean by 'posh', whether you mean 'upper class' or whether you mean 'affected.' In the United States, accent is much less of a class indicator than grammar -- not that the US has classes in the same sense as the British do.
There might be some in the US who say 'haitch,' but they would be few and far between, probably living in the backwoods of Appalachia. Right-speaking Americans would always say 'aitch' and 'sez' and 'pat-ee-o' and 'skedule' and 'guh-rahdj' or 'guh-rahzh'. And of course 'respite' is pronounced as though there were no e.
No doubt radio and television are responsible for some of the changes nowadays, but this phenomenon has been going on for centuries, witness the Great Vowel Shift.