I have been the first week in Jan. a couple of times and have had mixed results.
The first was my honeymoon (married Dec 30), and it was COLD. The sprinklers in the morning created ice on the grass, although it would warm up the the 60s by the afternoon.
The other two trips had very pleasant weather during the first week of Jan.
As far as crowds, I have found it to be very hit and miss. I think it depends on what day of the week the 1st falls on. As the first is on a Tues. some schools may not go back to school until the following week. Our best luck on crowds was when the first was a Sat, because most kids were in school on Mon the 3rd.
Another advantage is many holiday decorations are still up during at least part of that week.
I hope this helped instead of confused you.