Trail names generally do have to come from a government entity such as a park service, be it a city park service or the NPS. i.e. the ones who would put the trails on official maps and signpost/mark them. If we let any one individual or group informally invent the names for objects, then we blur the lines between fantasy mapping/vandalism and the on-the-ground rule and create loopholes for people to call anything what they chose to. The "widely recognized and used locally" rule (which is already subjective and contentious amongst mappers enough) would apply better if these weren't informal unsigned pathways on open public BLM lands, created, used by, and referred to by an already niche subset of individuals like local hobbyist groups such as a mountain bikers and hikers. The other issue with this logic is that, since no one entity controls the trails themselves because they are within BLM lands, any other group could theoretically come in and rename them under the basis "this is what we call them" just as well.

I hate to be the outsider to come in and mess with something you yourself are familiar with, but nobody owns anything they map or has personal control over it, especially if it's something based off a personal knowledge or a conflict of interest with a project rather than something that can be proven by anybody else with permanent signs, park maps, GIS programs, etc. You also cannot report people for simply following OSM rules that they are accustomed to following because you disagree with them. 

I created a thread in the tagging channel of the OSM US Slack server (https://openstreetmap.us/get-involved/slack/) regarding these trail names, asking whether the names should be removed or moved to a different name-related key as they are unofficial. While there doesn't seem to be a single clear answer, I'm more inclined to believe that from the responses from other editors, moving the names to a key like 'mtb:name', 'loc_name', 'nickname', 'alt_name', or removing the names wholly is more applicable than leaving them under the standard 'name' key.