Very true. There's a line of thought that the Pilgrims celebration was essentially their celebration of Sukkot.
Incidentally, there is speculation that the Gospel of Luke depicts the birth of Christ as having occured about that time. That would explain the crowding in Bethlehem, and of course, the shepards in the fields.
It should be noted that the brand of Christianity of the Puritans, namely strict Calvinism, more than any other Christian theology, most thoroughly integrates the Tanak (the Old Testament) with the New Testament. Essential to the Calvinism of the Puritans (and conservative Presbyterians today) is what is known as “Covenant Theology” the idea God works through a series of interlocking covenants with his people all throughout history. This seems to be exactly the way God related to Abraham, Isaac & Jacob, Moses and David too.
Covenantal Calvinists, like the Puritans were, will emphasize God’s covenant promise to David—that an heir would be the eternal King and Messiah, was fulfilled in Jesus. The New Testament book of Hebrews too, clearly says Christians are grafted into the tree of faith established by Abraham...and that we are his spiritual children and heirs as it were. The Puritan, very thorough bible scholars that they were, knew these things.
This may have as much to do with Puritan affinity with Jews than anything else.
The only people in America who have as much confidence in, and knowledge of, the bible as the Puritans did, are evangelicals...(especially the Calvinist ones)....hence, as someone said above, Southern evangelicals are, in a lot of ways, the “spiritual” heirs of New England Puritans.