Sure they did. Katharine Hepburn is one. If I recall correctly from her biography (or autobiography) she learned the pattern at Bryn Mawr.
Many people have very large homes – even compounds – in the Northeast, the Boston Brahmin accent, and lots of money, but that doesn’t mean that they are “old money.”
I found one website that lists the Kennedys as “old money,” but I can remember when they were not accepted in the old families. I suspect one reason is that they were not WASPS and they were fairly newly arrived.
I suppose Vanderbilt, Morgan, Reynolds, and Mellon would be recognizable names, but I read that many of the names would not be well known. Some of the “old money” families don’t even have the money any longer but are still accepted into high society.
I have found that a lot of moderately wealthy people adopt accents and become snooty. The wealthiest people I’ve known have been non-pretentious and down to earth. They have the dialects of the areas that they live in or grew up in.
I think that Safire is just poking fun without really knowing what he is talking about. Just a hunch.