I think Thanksgiving is really becoming the dominant secular American holiday that allows people of all cultures, faiths and religions (or lack thereof) to celebrate roughly the same general sentiment. It's probably helped immensely by the fact that the holiday was declared long after the the original experience was lost.
If we were to be true to the supposed history, we'd probably have to order some venison from the local Indian reservation, and make a variety of squash dishes. Or, if we were poor, we'd pile the table high with lobsters and complain that we weren't getting the good stuff.