It's amazing how we had any sort of industry before the car. I live in a relatively small (500,000) city, though Sweden's second largest. One of my clients is in an industrial area. The city has extra lines for the morning and evening commute, but only about 20% of the people use them, and I think another 5% cycle. I don't know how many walk into work, but clearly over 50% do drive in.
This in a city and country which is pro-mass transit, pro-bicycles and pro-high fuel tax. The turning point is often, as you say, when children are born. People do take their babies, in strollers, on the trams and busses, but it's much easier if you can leave spare diapers and such in the car and not lug them with you all day.
(Your comment about taking the bus with young kids to get groceries is specious. If density is high enough then it's easy to walk to a grocery store. In town there's three grocery stores within a few blocks of me, and a corner store .. on the corner. Surely that was the same in the dense European city you lived in?)
On the other hand, kids and people up to 20 ride free or at reduced price. This means greatly increased mobility for anyone over about 8 or so. I've read autobiographies of people raised in New York City during the 1950s or so, who describe the sense of freedom of being able to go to Coney Island, or Central Park. That's not possible without a car in the standard suburbia we're talking about.
That said, I really don't understand why a public transportation system needs "to be a break-even affair." The road system definitely is not. Gas taxes even in Sweden isn't enough to pay for the road network, snow cleaning, and so on. Public transit should properly be seen, like the roads, as having a multiplier effect. It means you can have areas of town with higher density, which support those people and families you are talking about, which have higher numbers of clubs and restaurants and shops, and have areas for families with cars.