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.

That's why you build separate bike lanes. It works fine for the Netherlands, but it does require a large investment in bicycle infrastructure.