OSM has a confused doctrine, visible on the ground out of one side of its mouth, and a strong focus on the English "public rights of way" out of the other.    In the US (at least in MA), public rights of way are verifiable at the registry of  deeds.   So I think access tags should follow what the actual legal situation is.   OSM also has a tradition of tagging signs, representing exactly what the sign said, whether or not what is on it is true, and whether the person who put it up is acting properly or not.

The case of Anthem Way is curious, and I'll follow up with the Acton Planning board.   It could be that L3 is wrong here, and it's easy enough to request the  subdivision plan, special permit, or comprehensive permit as those are all public records.