As uiri pointed out, transporting a family by bike is very common in other countries so it's not unrealistic at all. Though with how heavily the US is built around cars, the first wave of people trying it will have to make sure to live in the right places in the right cities.
Also, people who don't live the suburban car life typically don't buy "weekly" groceries.
It's actually usually a refrigerated van (bare in mind you'd have chilled and frozen items and those vans can be out for two hours at a time due to having multiple deliveries loaded).
But the type of vehicle is a moot point as it's not the shopper who has to own nor drive it. So they can still run a household without themselves owning a car (which was the point of the discussion)
You mean countries like Germany, Holland or Denmark?
I suggest you google for "fiets met kinderzitjes" (bike with child seats) to get examples of how you can transport 2 small children on one regular bike.
Once the children get older they can get their own bike. If you need to transport more than 2 children, get yourself a second parent or a bike that's designed to take cargo (yes I just called your children cargo).
My mother raised me and my sister on her own, using just a bike.
Seattle is a very hilly city, increasing the challenges for bike commuting for normal people. An electric bike can help but increases the price substantially.
At least in Amsterdam where I biked around a bit, it seemed flat as a pancake.
Also, people who don't live the suburban car life typically don't buy "weekly" groceries.