There's also this thing that even if you don't have kids (I personally don't) you don't want to sit on a car chair which has had other people's bodily fluids on it. Public transport like buses or trams is for the moment safe-ish when it comes to that, at least where I live, because not a lot of people masturbate in public or have sex on the chairs of a bus because people will call them out, but I don't see what would stop 2 people having sex in a car which they don't own (they only share it) and where nobody can see them.
There's also the non-politically correct question of what we should do with homeless people riding these vehicles? This is a real problem when it comes to public transport in my city, as the trams and buses are used as a de-facto day shelter against cold when winter comes. You really do not want to sit close to many of these homeless people in those cases, as the smell is very hard on the senses (I mean, real hard), often times you can tell a homeless people has been riding that tram half an hour ago only by the smell alone (even though he's not still on the tram). If said homeless people were to use this car-sharing thingie I can guarantee you that nobody else will use said cars after them.
There's also the thing of how do you protect these vehicles against destruction and negligence by its users, a problem encountered by many who have shared their car in the existing car-sharing projects.
That said it's a solvable problem with some thought, engineering, and standardization, but you are correct in saying it won't be solved until ride sharing EV's are out of the early adopter stage/commuter stage.
I don't have kids, so I'm definitely coming from a place of ignorance. On planes and trains I never see car seats used, which I assume is because crashes are so unlikely on those modes of transit.
If self driving cars are as safe as, or safer than, a trip on a train, would it still be a concern?
Not sure about trains.
I assume that if/when cars are as safe as a plane/train, adults wouldn't even need seat belts but you'd still need a carseat for an infant/toddler because it also stabilizes the child's position. Kids <6mo have no neck control. Kids <10mo can't reliably sit up for long durations. An autodriving car might be safe, but it'd probably still need to make emergency stops / turns. That'd be real bad if there were no carseats.
https://www.faa.gov/passengers/fly_children/
It's a PITA to use due to the huge variety of seat dimensions and car seat sizes and no standardized latching system for plane seats.
Before I had a baby, ride sharing was fantastic. Now it's basically useless. To transport my baby, I need a car seat of the appropriate size. It's really handy if I can leave a stash of diapers and wipes in the car. A portable high chair is nice, too. Sure, ride sharing is more convenient, but the car seat issue is a show stopper, at least until the ride sharing services catch up or autonomous car safety improves significantly and legislation catches up.
All that being said, the car seat and diaper cache in question live in an EV, and I'm quite happy that I am basically done with the gas station phase of my life.