Self driving electric cars that people pay for on a subscription model (rather than ownership) are the future I'd like to see. I hope it happens during my lifetime.
If the cars were self-driving, however, that would solve the majority of those problems. No need to hike to find the closest car, it would come to you. No need to pay for the time that it's parked while at your destination, it could be used by other people, and another car could come pick you up when you needed to leave. No need to return the car to where you picked it up, it would just go back to the closest available station and they'd rebalance themselves automatically.
Uber et al. have done a lot to help with this in my experience, but I've had many times where I'm in a location that taxis aren't typically in the immediate vicinity. Calling the cab company has often led to lots of waiting, not knowing when the car will get there, with cars sometimes never showing up at all. Knowing that the car is definitely coming to you (whether it's a self driving car, Uber, or any other service that tracks the vehicle's location) help remove that uncertainty and definitely makes it an acceptable option in more areas.
That reliability and knowledge justify the extra cost for taking Uber (or requesting a self driving car some day) to me, more than the comfort or style of a town car compared to a cab.
How much will the cost be reduced when we don't have to pay for human drivers anymore? Will we see 50$ a month (2013 dollars) plans for 50 hours of "chauffeuring"? (a quick google search shows that the average american spends ~40h driving a year)
So, the driver is a pretty substantial portion of that. Taking the driver out of the equation would substantially reduce the costs. If you increased medallion availability as well, and the owners were able to work with lower margins (or they were cooperatively owned), you could get the cost down even further, to maybe 1/3 of what taxis currently cost. That would be a huge difference in affordability for everyday use.
"That would be a huge difference in affordability for everyday use."
Very true.
Taxis drivers also really frustrating to deal with (the majority in my city, at least); I can't count the number of taxi drivers who haven't been able to find my house, so I've had to play phone tag with dispatch while waiting 45 minutes for a cab that should have been there in 10, then finally hike down the hill, get in, and had them start the meter while they drove up it to pick up my belongings.
But they have the advantage of actually existing. Whatever relatively near-term possibilities there may be for cars to operate on "autopilot" on certain highways, an urban environment where taxis are most common is probably one of the last places you'll see self-driving cars.
I think that I'll see them in my lifetime; I don't know if it'll be in 5 years (though I think that would be pushing it), 10 years, or 30 years, but I'm pretty sure they will come about, and will be a huge improvement over our current system of letting distracted, impatient people try to pilot cars through chaotic conditions.