If the reopened paths are permissive bridleways, as the tagging suggests, they might be better tagged as:

highway=bridleway
horse=permissive
bicycle=permissive
foot=permissive

If they're used by NT's estate vehicles, you could add motor_vehicle=private.

The access=yes tag isn't necessary and with the current tagging, would actually mean that horse-drawn vehicles have a legal right to use these paths (obviously it's unlikely that anyone would try this).

If you use highway=bridleway, highway=cycleway, or highway=footway instead of highway=path, these render as green, blue and red dotted lines respectively in the default OSM-Carto tiles.