Unless the signs were stolen, there are some very big yellow signs that indicate road is closed to winter traffic. That said, given the lack of snow and the use the road gets from snowmobiles in winter, it gets quite well packed from side to side and resembles a plowed road. In the past, prior to the base station road being plowed from the last condo road to the Mt Clinton road intersection a friend and I drove this stretch easily. One of the requirements for the state to start plowing the stretch of the base station road was that snowmobile trail had to be relocated off the road which required at least one new bridge. Jefferson Notch road on the other hand is frequently not gated due to snowmobile use. My first winter Jefferson attempt was started from the Caps Ridge parking lot on the day after New Years when the road was bare on both ends and could be driven by a regular passenger vehicle. This was a year where we had no significant snow until mid January. I expect the tow bill is going to be quite steep as they probably brought in a snowcat.
I have also seen winters where Sawyer River road was passable to the parking lot. I think the FS has made an attempt in recent years to just lock these roads once after hunting season and wait until spring to open them to avoid stranded cars. Then again years ago a hikers car got stuck on Caribou Valley road one winter and I think it sat there for several weeks until they could get it out.
It is unusual that the GPS did send them to Jefferson Notch road, my experience is that the logic usually defaults to primary highways. When I am going cross country over seasonal rural roads I frequently need to force the GPS to use these roads by setting intermediate waypoints at the end of these roads. At Baxter State Park, my GPS usually tries to get me to exit the park via the Abol Pond trail which hasn't been a road for 50 years.
No one hurt and they stayed with the vehicle, it could have been far worse if they had started walking. An embarrassment now but in few years a great family story.