First, we take this issue more seriously than most Americans. Most people shop and cook turkey, but do not engage in pilpul about mesorah of turkey and whether Yankees or Virginians started it.
2, it is misleading to connect the day directly to Pilgrims. Colonials/early Americans were practicing fasts and thanksgivings often on local and national level, more in the North than in the South and became national as part of Civil War.
Yom kavua that some objected here was started by FDR, before that it was up to the President to declare it annually. I am not sure why Nth Thursday is a yom kavua in calendar – seems like a variable day to me. July 4 is kavua.
FDR was prone to fight economy with magic, like Brandon, so he tried to move it to 3rd Thursday of November to extend holiday shopping season. This indicates that if there is any avoda zara in this day, then it is one of materialism.