Do families with babies and young children prefer public transport over cars in these places?

No reason why not. I see prams on public transit in Sydney all the time.

Depending on the age of the children they can also take public transport on their own, so can the drunk, elderly, and disabled.

(Kids are so overprotected in this age - freedom to travel on their own on bicycles or public transport is great!)

These people may not be able to drive cars. No, they can't get a self-driving car trip for $1.50 yet. When they can, we won't have enough road space anyway (one 8-carriage Sydney train carries 1200 people, that's a long line of cars!)

I really like what Citymapper is doing - augmenting buses with modern tech, similar to Uber Pool but coming from the other direction: https://medium.com/citymapper/good-bus-part-1-3-77d65e6f8ce3

To an extent, yes. Basically you have three possibilities: a) daycare is close to home, b) daycare is close to work, c) daycare is in random place.

Of these, b) is the least common, but for these people public transport works well. From our previous house, we did this, I took both kids (5 and 2) by bus with a jogging stroller to and from daycare every day. This was approx. 15 minutes faster than going by car, due to dense traffic and very limited parking spaces. Now we live close enough that we bicycle instead.a

I mean if you asked them, they might prefer a helicopter ride or a flying car, but they certainly do use buses and trains. Subway stations without elevators or escalators are not ideal for strollers though.

None of the people in my gf's family in singapore has ever owned a car or a driver's license.

Yes, even here in a rural area with good public transport. Not just us as parents, but our 2-year old loves going by train or bus.

yes

Yes.