I haven't thought about this for quite some time, but I remember the local mass transit, DART, offered shuttle vans if people got together and showed enough interest in people meeting in one spot and being dropped off in one spot. DART provided the driver and van, and the users just paid whatever the fare. This allowed DART to offer service and acted as a trial run on if a full bus route was needed.
Seems like something that whatever transit authority can use as well. Uber just has a better PR department with much larger budgets than metro agencies, so to younger people this probably seems like an original idea.???