People signed a document with all sorts of religious references and prayers in it on July 4 1776, citing Divine authority as their sole justification, so that has as much claim to religious origin as anything about Thanksgiving. Thanksgiving is more of a secular religious holiday, rather than possessing any religious significance in its own right. It's more historical, such as if the Church had the guts to celebrate the anniversary of the liberation of Jerusalem by the First Crusade. A secular remembrance of a religious event, made into a legal institution for commercial reasons. You can't water down a religious holiday any more than that.

Also, Christmas is not much of a religious holiday, it is almost like two coincidental celebrations, one cultural (I know of Jews who do the tree & gifts thing who wouldn't be caught dead in a church, and would not sully Channukah with practices carried over from that gentile custom) and the other religious, and the two kinds of observance are seldom intertwined. The government does not recognize Christmas as a holiday because they honor the birth of Jesus, they recognize it as a holiday because nobody would get any work done if they did come in, and they all want to be home with their families anyway. The legalization of the Christmas holiday was more of a de jure recognition of a de facto institution.

By the way, New Years Day is a Holy Day of Obligation for Roman Catholics in the USA. It is the celebration of the Feast of the Circumcision, because it falls one week from Christmas. Both the observance of the Nativity and the Circumcision have absolutely nothing to do with the actual contemporary practices of exchanging gifts and celebrating the changing of the calender, and to the extent that the secular practices have any religious origins, they have become so divorced from those origins that said origins no longer have any meaning for the practitioners.

My family exchanges gifts and meets for a meal on December 25, because we are Americans. We go to Mass at midnight earlier in that morning, following four weeks of penitential observance, because we are Roman Catholic. If we changed religions all at once, we would still do the former. If we ended up in a situation or place where the former was not practicable, we would still observe the latter religious practices. Really, I think of the two aspects as distinct from one another, which is why I don't pay any attention to the War on Christmas stuff, and in practice, most other people do so as well.

Thanksgiving? Not even on the radar. That has absolutely no religious context for me, and people like me or anyone I have ever seen that is not a TV character (and usually not on a good show - you never see 24 or The Wire characters going around the table saying what they are thankful for). In fact, I would say Independence Day has more of a religious connotation for me, because my political views are based on what I believe "the laws of Nature's God entitle" people to have.