Puritan Board Doctor
I'm kind of surprised at the number of Reformed brothers and sisters (generally speaking and not limited to PB) who are fine with remembering the secular reasons for various seasons and even partake in them, but are strongly against acknowledging the religious aspects be it at home or in the church.
Christmas has nothing to do with Christ's birth since Scripture does not command us to celebrate this day as a holy day for his birth. Since Scripture does not command us to celebrate it for his birth, then to do so goes against Scripture. It doesn't matter who turned it from a pagan holiday into a "Christian" holiday God didn't and that is what matters.
As for a holy day or religious holiday, God doesn't look kindly on it any more than he looked kindly on the Israelites for forming a golden calf to portray who he was. We can't just do whatever we want because we think it's righteous. The Israelites had become used to worshipping idols in Egypt. They thought they could place God into a golden calf and worship him by bowing down to it. They probably thought they were doing a righteous act too. God didn't give us a command to set aside one day out of the year to celebrate Christ's birth. A religious Christmas is a manmade holy day ill-conceived by sinful man.
As for its pagan roots, that's just what it is "roots" which no one but actual pagans celebrates. They actually call it something different and I doubt it falls on the 25th each year. I don't know much about it. If its roots bother some people, I can respect that and would advise them not to celebrate Christmas. I have no such qualms over its pagan roots because I don't associate it with paganism. I associate Christmas with my family's rituals or customs whatever you want to call lit.
As for a secular holiday, if we decide we can't celebrate the Christmas holiday in a secular fashion, then you will have to apply that command to every holiday. Although, I'm not sure which Scripture you would use to enforce this command.