There seem to be few hundred uses of surface=bare_rock. It’s not documented on the wiki, though.
Is that typo or not? If not: no, I would not mark surface of highway=path
with natural=bare_rock, as that one is documented to applying only to areas, not ways.
If it was a typo, and you meant surface=bare_rock
, I would consider it once it is documented on the wiki, and its description matches what I see. But as that one is currently undocumented, I would currently avoid it and choose best matching documented alternative (likely surface=rock).