So we have two young kids and a car. I am of the mind that we could be car free, but my wife appreciates the convenience of having one. We have a lot of experience with Uber and kids though because we travel a lot and rarely rent cars if we're traveling to cities.
We live in Chicago, so a dense urban area with good public transit in the city and basically no non-car ways to get to the suburbs. We both do entirely public transportation commutes, and we walk the kids to/from preschool and daycare. We use our car once or twice a week - for the grocery store (which can be done by transit but I'm much more likely to go to Aldi if I use the car), and to drive out to suburban locales to hike. The hiking is actually the biggest reason we don't give up the car. It'd be limiting, not to mention tiring, to be limited to places we could use a combination of transit and bikes to get to. That said, I think a few ZipCars a month would probably be more efficient than car ownership.
The thing my wife and I really disagree on is how much of a time penalty we're willing to endure to take transit. And she has a point - it's sort of the worst to have a cranky kid and not be able to make a quick exit to the car. Instead you might be facing 30-45 minutes on the bus or train to get home, just HOPING they fall asleep in the stroller.
The hardest thing about Uber with two car seats is fitting in both adults. If all four of us are Ubering, we often take two Ubers, which of course doubles the cost. But the alternative is to squish one of us between two convertible car seats, which is super uncomfortable and seems at least somewhat unsafe for everyone. But that said, the installs/child wrangling is much easier when we are both traveling.
A seat belt install with an infant bucket is pretty easy - you don't install the base, you do a seatbelt install once the child is already securely strapped into the seat. This means you can meet the Uber driver carrying both seats, place them into the car and then install each one while the children are contained inside the car.
I'd try it out when everyone is in infant buckets and see if you have a desire to continue once it gets a bit harder with convertible car seats. But just to give you a mental picture of how solo Ubering works when I'm with my 1 year old and 3 year old.
1. Put the two car seats and their bags on the curb before I call the Uber.
2. Put the three year old in a carrier on my back. Call the Uber and carry the one year old in my arms to meet the Uber.
3. Seat belt the one year old on the drivers side/street side of the rear seat.
4. While he's relatively contained by the seat belt, install one convertible on the passenger side, then move him into it.
5. Then install the other convertible on the driver's side/street side, and put the three year old into it. The worst part of this is you're on the street side with a child on your back, which is probably not the safest. This can be replaced by having the kiddo in a stroller, except my three year old can houdini out of a stroller.
6. Get in the passenger seat, apologize profusely to Uber driver and assure him he'll get a good tip.
Some of these shenanigans are because my three year old won't just sit in the car while I wait to install her car seat, so she has to be contained somehow until I can strap her into the car seat. You are right that a lot of this probably is solved by using ZipCar. However, you will just need to become a car seat installation expert because an improperly installed car seat is no bueno.
That said, if you are dedicated to being car free there are some alternatives out there to know about. The Immi Go is a foldable convertible seat that folds down to fit in a backpack and weighs 10 lbs. It requires the top tether but all modern U.S. cars should have the tether available. It is for children over 1 and 22 lbs, but note that it is front facing only so you should wait until at least 2 by best practice. The Ride Safer Travel Vest also works for children who are at least 3 and 30 lbs. We're thinking about getting one because it would help solve the "can't fit the second adult" problem in Ubers. Finally, the Cosco Scenera is an excellent travel convertible seat - 7 lbs, easy to install rear facing, and low profile enough that I don't have to ask the driver to move their seat up to install it.