I'm not sure why the should "operate" anything. Any taxi or Uber driver could autonomously decide to put up a route sign and start following that route, with a standard ticket price that makes the service profitable.
Overwhelmingly however it's cheaper to vertically integrate, and private operators have no interest in taking low profitability routes (which can often be very important due to second order effects).
I will contend that automated busses might change things here a bit though.
No. The public transport authority keeps doing exactly the same that it's doing now. Simply, taxi drivers can choose daily to start following a route for shared drives. Nothing else, except maybe some coordination so that the ticket price is known in advance.
On a smaller bus line with less frequency than that, it will also not be really profitable for "independent" drivers.
It may be useful as a temporary solution or a local test but a public transport authority (should) have enough data to scale lines or create routes based on real usage.
When public transport are bad, it's rarelly due to the physcal constraints but always because budget is lacking. You aren't going to solve your lack of bus (drivers) by adding more vehicules with less capacity.