I have two kids, not three, but we have never had a car, always taken the kids everywhere on public transport. I’m not going to say every trip was a joyful experience, but imho you are making a way bigger deal out of this than the reality. Although to be fair it depends a lot on the quality of the public options in your metro region.
Downside of kids that always travel by public transport: on the rare occasions that they do travel by car, they puke nearly every time. For us public transport is a lot less stressful than traveling by car.