Taxis are not the problem. Private cars are the problem, and the city does a pretty good job keeping them out of Manhattan as well, with tolls on many of the bridges (any bridge into Manhattan connected to a highway is tolled; local street bridges are not) and a 19% parking tax.
But there are certain situations where you need a taxi and the bus or subway won't do. Think the elderly, families with small children, etc.
I would love to see a city mandate something the size of the Lit Motors vehicle for the cases you mentioned. However for many of the cases you mentioned, taxis still aren't the solution. There are plenty of families and handicapped people that get by with public transportation. They can use things such as scooters, wheelchairs, motorized wheelchairs and strollers. Technology has advanced to the point where a motorized wheelchair or scooter is no longer a rare luxury.
While I can see the argument for taxies, I;m totally in agreement about private cars and especially parking. Parking has no business in the modern urban dense city.
But there are certain situations where you need a taxi and the bus or subway won't do. Think the elderly, families with small children, etc.