This path has been set to highway=no access=no by the NT team because
Path access has been set to =no due to the fragile nature of the area and ongoing peatland restoration.
However this path lies on access land which means there is a right to walk on it. Should this be set instead to foot=discouraged? If there are signs or barriers that discourage walking there?
Both sides of Snake Road (and especially to the south near Kinder Scout) there are lots and lots of paths across the moorland. Someone’s even just added a note above the ledge to the east saying exactly that.
Technically speaking, you’re entirely correct that foot=yes applies anywhere across that moorland unless some sort of restriction has been put in place (which might have happened here - you’d need to check the signage or ask the NT). Personally I’d have gone with disused:highway=path rather than highway=no (and did, when I mapped something similar in the NY Moors a few years ago).
If people avoid the closure by heading east toward https://www.openstreetmap.org/way/143220750 and then north around the closure, and over time enough people do that so that it’s verifiable as a path, then that would also be foot=yes, because as you say it’s on access land.
However, the edit was 3 years ago - it’s likely things have moved on a but since then. The mapper who edited it hasn’t edited in a couple of years, so maybe a changeset comment wouldn’t help.
I’ve seen similar on Dartmoor. Paths blocked because of damage. My view was the action does not conflict with right of access. The right is to be present and walk across the land, and not to use a specific feature on the land (eg path). The land owner can remove the path by making unsuitable for walking, but they can not fence off the area of the path. On Dartmoor this involved putting obstructions along the paths with temporary signs explaining it was to protect habitat and asking public to use nearby paths.
Note that NT’s own tagging reference guidelines, formed in collaboration with the OSM community, recommend that they map using *=no values instead of *=discouraged - a comment in the document indicates that that was due to the limited usage of discouraged in practice.
Ref: File:National Trust - OSM Path Schema.pdf - OpenStreetMap Wiki