There are bikes that would work for you. I think you have to decide that it's okay to spend some money on the right bike(s) to make it work. Once you accept that the bike is your "car" and you can spend the kind of money on a bike that you would on a car (not as much, really, of course) you are then free to find the perfect solutions regardless of cost. In the end it still costs less than a car and saves even more over time. You would probably love a fat tire/snow tire cargo style bike for those long treks to the grocery store, maybe even with a nice electric assist. Then something lighter and faster for your other situations. And, of course, you need the proper clothing as well to make it comfortable in inclement weather - it can be, really, just look at the folks in Copenhagen.
I did it in stages. Got the first bike and used it whenever I could and found the situations that still needed solutions. Got the second bike and the trailer and went a few months with no more issues, then I sold the car. I wanted to be pretty sure it would work out without regrets. And I don't have any. Of course, I live in an area with fairly good public transit and also milder winter. Ymmv.