No. With paths that are hiking trails, based on my observation, access tag is wildly misused to also indicate practical usage. To be precise, I should have used horse_scale= that I did not even know existed. However, the main distinction between different paths/pathways should not be based on legal access (usage in your parlance? - though I can see you use the word in two meanings: 1) practical usage 2) legal usage)
but practical usability=what it physically looks like.
Sidenote: I find the winking smiley mildy passively-aggresive. Well, it is more economical, it is duck-tagging, so infinitely easier to grasp, it is easier to parse for renderers, it would solve the path controversy=ambiguity of nobody having no idea what path is.
As for access tags, not really, shared_use would have foot and bicycle implicit (and all the others probably too dependent on the region, co motorcycle=no, horse=yes, but that can vary regionally).
For the ways that are not signed and physically allow comfortable usage of motorcycles, biked and what not, we could have either path(way)=motorcycleway/multi_use_path. However, for trails (i.e. ways not really suitable for normal motrocycles and bikes, which I think are the majority of paths now, one would use highway=trail, with appropriate difficulties scales to add suitability for different modes of transport. Again, you ae fixated on legal access, but I do not see a reason why that should form the criteria to base division of highway=path on. Shared_use being the exception here, but with a big practical usability caveat.