If you want a car at the precise time everybody else happens to be commuting and commuters are the principal users of on-demand car services, it doesn't really matter whether your commute is shorter than some of the others. It's one extra car needed to be available every day at the time when all the other cars happen to be in use and (like most of the rest of the cars in the commuter pool) it'll spend the rest of the day sat in a parking spot, costing little in mileage but plenty in depreciation.
Of course if commuters needing to be around for the start of a normal work day aren't the principal user of the service, then cars can be used more efficiently throughout the day, but then they're not "competing with private car ownership" to any significant degree.
For edge cases like you whose car use consists of occasional ten minute drives, it might already be cheaper to just take taxis/Ubers
Like everything - high bandwidth vehicles will do the bulk of the work, and seamlessly trade off to "last mile" self driving small vehicles.
If the car is electric, it isn't so bad on the environment, and it can park itself in a no humans allowed garage (so little chance for breakins). As for rush hour, I'd hope swarm scheduling lowers congestion substantially (though this and making longer commutes tolerable really just means I'll live further from work).
Still doesn't solve the problem - just moves it. If everyone in my office complex needs a ride to the bus around 5pm, and everyone at the other end of the bus line needs a ride home at 5:42pm when the bus arrives in SuburbTown, there will still be unmanageable shortages of self driving small vehicles - unless you own your own.
Huh? If suburban Americans aren't opting to share rides into work now and don't have or want bus services now, particularly not connecting bus services, what difference is non-human drivers going to make?