Shared-use paths work best when they’re low use and low-speed. Ergo why people will walk, bike and drive in the road on a cul de sac but not on a main stroad.
It’s common to have separate sidewalks and bike paths on faster, more commonly used routes, because bikes don’t actually mix all that well with pedestrians. It’s the same reason we don’t make sidewalks wide enough to drive a bus down.
By your logic, public car roads are fine so long as there’s a bus that drives down them. Even if 99% of the people on them are in a privately-owned bike or car.