As far as holidays go, no Christian is worshiping Tammuz or Ishtar. They repurposed holidays within their own Greco-Roman culture as evangelistic tools, and the tradition stuck. Given that Psalm 29 was originally a hymn of praise to Ba'al that David repurposed to praise the Lord instead and Paul repurposed an alter "to an unknown god" in Athens as the jumping-off point for preaching about the true God, I don't think that's a sin in and of itself.
The real problem is that by rejecting the Biblical Feasts, the Church both lost some of the most potent symbolism pointing to Messiah and intentionally rejected any association with its Jewish roots. In the process, they took a body that was supposed to be "neither Jew nor Greek" and turned it into "Greeks only; Jews need not apply."
Shalom
The “alien in the land” didn’t seek the benefit of the oracles of Yehova. He was lost.
As for Xmas and Ishtar, the original effort by Constantine was the defeat of the Gospel, not the repurposing of his pagan worship. He persecuted those that stuck to the Way. He slaughtered Rabbis and scribes, and he burned manuscripts.