As a parent myself who has always had a car but has aspired at times to go car-free, I think the whole "semi-frequent short term car rental" thing only really gets realistic when the kids are old enough to just need those tiny Mifold booster seats / seatbelt adjusters.
Conventional UAS-attached car seats are just too bulky and troublesome to be installing and uninstalling all the time in cars you only rent for a few hours at a time.
I think the key there is "all the time" - when we lived in the middle of a European city we used car-share, and the car seat was a bit annoying but since we were using it maybe once or twice a month tops it just wasn't that big a deal. Though we were using the seatbelt mount, and not ISOfix.
I looked into carseat rentals about ten years ago, and it sounded like it's only really common at the agencies located around airports. Overall, I got the sense that there's a lot of fear and liability tied up in it— that giving someone the wrong seat or installing it incorrectly or not cleaning it adequately were all potential landmines waiting to happen.
Conventional UAS-attached car seats are just too bulky and troublesome to be installing and uninstalling all the time in cars you only rent for a few hours at a time.