The snowmobile=* tag is an access tag, and should be probably be used with a highway=* tag to cleraify what kind of road it is.
The path_example is just an example. There is no indication on https://wiki.openstreetmap.org/wiki/Key:snowmobile thta it should only be used with paths.
The history is that I did not setup the orignal track as unclassified. The thing I did was to handle the intersection with the E4 highway.
