This is probably worth a broader discussion that isn’t limited to the U.S. Previous discussions about street relations have run into a familiar slippery slope argument, that an overzealous mapper might blanket a city in street relations, making any kind of street editing a royal pain. Indeed, this has happened anyways in some cities using route=road, in the absence of a dedicated tag for streets. On the other hand, fears of relation overarchitecting haven’t borne out with either route=railway or type=waterway.
Between that and the community’s apparent acquiescence to route=fitness_trail, a tag to distinguish non-fitness trail relations from trail route relations would seem quite logical to me. That said, I’m not sure it would clarify the issue at hand, about naming the member ways. No one would feel comfortable deleting names from roadways just because they’re members of street relations.