Where I come from, big city etc. there seems to be not much time advantage of taking a car vs. taking a bike. After all a bike is very easy to park close to your destination and it's not affected as much by traffic jam (due to rudimentary cycling infrastructure and the ability to use side streets that are too narrow for rush hour drivers to consider).

WRT child transport: child transport bikes are seeing a rising adoption here, so it is feasible to bike to work and drop of your kid on the way. Kids will stay dry below the bike's rain cover, though of course the parent who cycles will need some upgrade to his clothing compared to what's needed when going by car (finding cloth that is both rain-tight but doesn't cause sweating takes some experimentation).

If you want to look at the spectrum of possibilities, have a look at e.g.

https://www.babboe.com/

https://www.bakfiets.nl/

https://www.urbanarrow.com/

http://www.nihola.com/home.html

https://yubabikes.com/

(just enumerating what bikes I see on a regular basis, this list would go on and on)

PS: you may also want to factor in the time spent on maintenance and bureaucracy (dealing with car insurance etc) and the extra monetary effort involved with owning a car vs owning a bike.