But, the only alternative considered is private individual/family transport.
Why is private mass transport not more widely available given that it can solve a lot of these problems?
Having an Uber for buses which does smart scheduling based on current demand, possibly involving transfers so that frequent local routes connect with each other without long delays, should be possible.
Of course, prices will fall when things scale. So, the government can be involved as a facilitator but operations are mostly run by companies which can pay a fee to the government rent necessary infrastructure.
You still have the problem of higher prices for odd hours/locations but sharing costs ahould make it cheaper than uber.