I think the Romans chose January 1st because it follows so soon after the Winter Solstice and instinctively people knew the winter was coming to an end. The shortest day of the year passes as the Mother Goddess gives birth to her Sun (son) when it's first rays touched the altar stone at Stonehenge. It felt as though there was hope to get through the rest of the winter and a time to start over.
While the Sun is just a babe, the winter belongs to the Crone or Cailleach. Hers is the power of death, the dark, secrets, seeds, and all that is wild and unborn. Nothing is beyond her reach. The three moons that occur in this season (Jan-Mar) take us from Winter into Spring, but it's the Esbats, the dark moons, that are for dedicating yourself to your magical universe. Use these new moons to carry out a dedication ritual, stretching it into all three months. Recognize what you have been and are choosing to be. What will you dedicate yourself to this year? To what will you give birth this spring?
The most powerful among us take responsibility for what we create and do so with intent. Be aware of what you create for it effects us all.