I worked with a guy who had two kids and no car. They rode their bikes everywhere, always. He insisted on it. I think his wife was miserable lol. But his kids seemed to like it!
Rain or snow doesn't really matter either; that's what fenders and a jacket are for, and in any case it rarely rains heavily non-stop, but comes in waves, so I can just wait until the rain pauses for a little while.
It's great fun, actually. I hate getting groceries by car now.
The worst part is that the shopping carts aren't near the bike rack, so its quite a hastle to deal with toddlers trying to run away.
It's 100% about the equipment. Gore-Tex. Neoprene. Fenders. Front and rear racks. Waterproof panniers. Lithium-ion batteries. Motor.
Park right in front of the market. Pop the panniers into a cart. Stow the groceries directly into them. No hauling bags out to your car, loading them into the trunk, and then unloading them at home. Instead, transfer straight from the panniers to the fridge and pantry.
I do it year-round in the hilly and rainy Pacific Northwest, and it's actually kind of fun.