Uber and Lyft may well be 10 times the price public transport, but the services are otherwise barely comparable. It absolutely is worth $20 to me versus $2 to get where I'm going in a fast, comfortable, and private manner, in exactly the same way as I don't baulk at a $3 Starbucks, despite being able to brew a coffee myself for 1/20th that price.
Public transport is never going to win simply by being a cheaper option, you have to appeal to other motivators, such as sense of social responsibility.
As an aside, my "future city" vision of public transport would be publicly available, self-driving, electric "cars" recharged with renewable energy. Not mass-transit. I don't want to be forced into a shared space with strangers, if I can avoid it, thanks.