I just don't see that happening. After all, people outside of cities can use taxis right now, yet they do not, for many good reasons, including cost and convenience. Autonomous will simply be another option for the family truckster.
May I ask if if you drive, and whether you live in a city or not?
No way. Obviously you're not a parent. Carrying around a car seat (or several!), so that you'll be ready for your next "auto-Uber" ride ain't gonna happen. So there'll need to be child seats built into every public vehicle, or at least a good subset, and their existence will have to be a call ahead parameter.
I live in major cities. There's no question that my focus is to the effect on population centers and rural experiences will be quite different but rural populations are also shrinking. The 2010 census had urban population at about 80% (and continuing to increase every census). To be fair I think there's a higher population density than just "not rural" needed to support some of these concepts so let's exclude urban centers from the definition and just focus on urban (by census bureau standards "urban" is larger than "urban center") which still leaves us a little over 70% of the population. And that will only increase.
I do drive, in fact my partner and I have multiple vehicles, more than we need as some are just "fun." I like cars, I like driving, I also realize that I am (willfully) throwing away a large sum of money on depreciation (even more so because I buy new) just for the privilege of these big pieces of equipment to sit in my garage going largely unused. If we look at usage across all of my vehicles they are maybe used 5% of the time. I'm using an hour of drive time per day and not accounting for time I spend traveling (as in out of town) and that's across all my vehicles not "each". That's a very generous number and likely inflated, I would not be surprised at all to find the real number is more like 1-2%. Further, that 5% is mostly spent on uninspiring transportation, e.g. sitting in traffic or driving to a restaurant or activity, heading on an errand or to the gym. These are not "fun" drives by any means. I could easily see myself subscribing to a TaaS provider with a decent SLA (i.e. guaranteed to arrive within 8-10 minutes max). With that sort of timing I can make spur of the moment decisions to go somewhere at will and the ride will arrive by the time I've gathered my things and put on my shoes.
I don't think comparisons to taxis are apt. Taxi service has always been subpar. You need to make a phone call (outside of cities that is) and a dispatcher will tell you that the ride will arrive in 40 minutes which may or may not happen. The prices are inflated due to artificial constrictions on service with government limits on medallions. That same constriction has allowed providers to deliver horrible service without fear of loss of business. All of that is coming home to roost with ride sharing now eating their lunch although you still do have the variability of drivers in the ride sharing sector (some try to screen out rides they don't want or accept the ride and then take 10 minutes to leave wherever they are and start their journey to pick you up). With the elimination of the human factor that gets even better still. Again, leave the rural areas out of the discussion for a moment because the lower density makes ubiquitous availability challenging but within a city or suburb big data can make TaaS readily available within a reasonable SLA to everyone and for far less than the cost of owning your own vehicle. That's a compelling situation and I expect a big majority to at least reduce their own vehicles (e.g. single car household rather than two or more) and at least a simple majority or urban dwellers to cease owning a vehicle all together. I feel I'm being conservative with those statements but time will tell.
I already make enough use of Uber that a monthly flat rate would make sense for me. I use it whenever I travel - which is frequent - and even when at home if I want to have a cocktail or if I know the parking situation will be challenging and no valet exists. Make that autonomous and offer TaaS packages and I'll definitely be on board.
As to parents, I'm confident of a solution. Maybe first time parents and helicopter parents will be in the subgroup that continues to buy their own vehicles outright. Perhaps there's a way to collapse infant and toddler car seats into the vehicle itself and pulling a lever or clicking a button will make it available. I'm sure there are many ways to solve that issue. People that have gotten over the "oh my God my baby is so fragile everything must be 100% perfect and sterile" won't have a problem with that (news flash, babies are about as far from sterile as it gets, they are little virus incubators that share their pathogens with all the other babies they come in contact with and spread culture samples across every surface they touch

good thing for them they are cute). Off topic but the Freakonomics guys did a good piece about the efficacy of car seats beyond the youngest ages; spoiler, they're actually more likely to cause harm in some types of crashes and fair no better in any type. Tying it back in though, a problem with child car seats is that they are not designed in concert with the vehicle manufacturer, having an in-built option could even be a step in the right direction.
People have highly emotional attachments to vehicles, I'll include myself in that. That's changing though, younger generations are much less likely to see driving as a necessity and just view transportation as utility rather than activity. The romantic notion of the vehicle being gateway to freedom and expression is on the decline. I think this transition will happen much more quickly than most people expect.
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Plus neither taxis nor Uber even exist where I am, and I'm only thirty miles west of NYC
Depending on the city, that's already happened. Heck, I know a lot of native New Yorkers who never had a driver's license.
See, that's where I disagree with you guys. As a rural or even suburban dweller, I sure as heck am not going to wait even five minutes for a public vehicle to show up, when I can jump in a private vehicle and go... with it right there waiting for me when I finished whatever I'm doing. The convenience is just too great to give up.
E.g. Christmas shopping would become a nightmare if I couldn't store stuff in my car between trips. Ditto for every day trips carrying the kids and their friends.
So unless it becomes far cheaper... and I can't see how that would happen to normal people... I just can't see a mass switch away from privately owned cars. It didn't happen with public transportation and it's not happening just because cars will be able to drive themselves.
I think even my mother and my wife, neither of which can drive any more, would far prefer to buy their own autonomous vehicles.
Well, we'll find out in a few years what the trend will be. Should be interesting!
To be fair 30 miles outside of NYC is the boondocks, people don't realize that unless they've gone upstate but there is wilderness just outside the city. I was planning a trip with a friend upstate once and after realizing there were no major hotels in the area I went to AirBnB and I literally found someone listing a chicken coop. Upstate and NYC are two different worlds.
Shopping, we already have the option to purchase (even in brick and mortar) and have things delivered to our homes. That could easily expand. When I'm doing a lot of shopping I don't want to bring everything to my vehicle and into my home already, I have it delivered.