Bingo! The NFL wants games with slots on the line to be played at the same time.
Ergo, if the Saints win tonight -> they play at 12noon on 1/1, same as St. Louis-San Fran, with the #2 seed on the line.
If the Saints lose tonight --> they play at 3pm on 1/1, same as Atl-TB, with the #3 seed on the line.
So no rest for the Saints and Brees.
If it makes anyone feel better, even if the Saints had beaten the Rams and/or Bucs, there would still be no rest for Brees & Co., because the #1 (if both won) and/or #2 seeds would be on the line in week 17 (depending on the tiebreaker with SF). Kind of a weird year that way.