Most car seats I've seen are difficult to transport, mainly because they are heavy, bulky, don’t fit onto a stroller, and they certainly don’t roll. Once I found out Uber had that feature in New York I didn't even bother doing additional research for that trip, but could benefit from the knowledge in future trips.
We've done two plane trips with her so far, and obviously the car seat has come with every time, and it's worked out very well.
I'm not sure whether the car seat is heavy or not - just earlier, I was discussing laptop weight with someone, and I definitely don't mind carrying heavy things - so I honestly don't know if I'd call it heavy or not. It feels fine to me, and if I had to take her somewhere in a car without the stroller, it would be annoying, but much less annoying than owning a car in the city :P
I also don't really see why you necessarily have to limit yourself to one kid; there are plenty of taxis that aren't small sedans, and most Uberlike apps allow you to sit in the front, so you could pop two kids and a spouse in the back, and ride up front.