The scene when Roger Wakefield arrives in Boston in a Checker cab. The Checker cab has the federally mandated 5 MPH bumpers which was not until 1973. The scene is dated to 1968 or 1969.
Brianna tells Claire she looks like Jamuhie so Frank must have hated her. Claire and Jamie both have blue eyes - it's a statistical impossibility for two blue-eyed parents to produce a brown-eyed child like Brianna.
As the live TV broadcast from Apollo 8 wraps up, one can hear mission control saying "three minutes to L.O.S" L.O.S. means "loss of signal". which is what happens when a spacecraft crosses over to the far side of the moon at which time line-of-sight communication signals are blocked by the moon itself. Yet, the live broadcast shows the earth in the live image. The broadcast happened not long after the spacecraft emerged from the far side, and not just minutes before signal loss. The loss of signal for each orbit was about 45 minutes, leaving about 1 hour and 15 minutes for communications each time.
The Charlie Brown Christmas TV movie had only appeared three times before 1968. It wouldn't become much of a tradition for several years to come. When it debuted in 1965 Briana would have been a college freshman, which was a bit old for it to have been of particular interest for her, much less become a family tradition.