The cargo bike boom has brought us some really decent ways to transport small children and stuff by bike, I actually think it's quite possible to use is you live in a not so small town. There are accessories to weather proof the cargo area, there are Iso-fix mounts for child seats and once the child can sit by itself it's usually quite a joy for them as well. These bikes are also protecting the child in case of a fall much better than you would think.

However I really do understand that a car is significantly more convenient. I live in rural Germany and there distances can easily amount to 10-15km one way to run errands such as going to the pediatrician. It's just a bit much, particularly with a toddler. And the car really does become storage for clothes and all that, you can just park it and everything in there is dry and safe, all that makes the car very attractive. Also a decent cargo bike with kids-friendly accessories will run you as much as a cheap small used car, although only the initial cost of course.

The key to bad weather is decent clothes, and children can easily be weather proofed for the most part. My kid is three now and I've seriously considered switching over to a bike, but only to replace the second car that I frequently use because my partner will need one to go to work anyways. But running the car cost me around 250€ every month (I keep track of every expense except cleaning) and that is only as long as nothing major breaks. Upkeep of even a large cargo bike is a fraction of that.