It sounds like a bike is not for you, but for those who use a bike, they do have solutions to your problems:
Weather: Cold is easy enough, unless there's deep snow. You buy warm clothing and take it on before riding and off afterwards. Those clothes are also perfect for taking the kids for a walk, hitting up playgrounds/forests. Rain can sometimes be fixed by being a bit flexible with when you set off, else waterproof clothing. Hot weather is probably harder, although you do get some air cooling once you're on the bike.
Shopping: Actually, with a child seat, you can fit a relatively large bag of groceries, enough for a couple of days.
Family: You outfit your partner with a bike too, and the oldest kids. The small ones (say, under 5 years) you put in child seats on the back of the bike or in a trailer. Range is of course limited, you can't go somewhere far away.
If you think all of this sounds uncomfortable, one really important upside is that riding a bike most days makes you more active and arguably tougher, both of which are important traits with modern sedentary life styles.