I am 100% for reducing the amount of cars on the road but this logic doesn't really work when you've driven in from out of town and there aren't sufficient parking garages.
And all the families that currently live high quality loves, without cars, in these places would find your assertion about “can’t have a family without cars” absolutely comical.
For large chunks of the US, top down urban planning has forces people in most areas to own cars. But it doesn’t have to be that way, it was a choice that was made, and a choice that can be changed.
https://priceonomics.com/how-the-world-shops-for-groceries/
I live in NA, but a grocery store is an easy walk, and was on my daily commute in pre-COVID times. This makes me an outlier among North Americans I guess. But it would be the norm for places that allow for car free life.
The following is therefore speculation, but the best I can come up with to try to meet you where you are. Right now, your car based life is so inconvenient and so inefficient, that stopping by a store is something to be minimized, to do as infrequently as possible. So to you, going to the store more than once a week is an inconvenience. You have to drive somewhere special outside of your normal route, park, walk a long distance from the car to the store, wander around a store so large that the idea of stopping and grabbing something quick is almost inconceivable. You likely even have to wait in line, because checking out a single person with a full week of groceries takes many minutes.
Once we allow the possibility of not living with a car, which means all of your daily needs within walking distance, an entirely new world opens up that is much more convenient. There is a smaller, well stocked store along most routes you take. Grabbing groceries for a day or two is not an affair that has an extra half hour of driving and walking. It's a quick hop in and out. You can always have fresh food, because you stop frequently and quickly, without having to wait a week in between trips because to shop more frequently would be wasteful of time. Run out of sugar? No big deal, it's conveniwnt to buy some more.
Once cars are gone and cities are designed around that, everything is more accessible and convenient. I mean, the whole thread started with how life in cities so not need cars, and how having cars in cities ruins life there. And for that matter, much of small town life can be similar to cities, because there, needs can also be within walking distance for a small compact town.
It's only car-based suburbia with highly restricted unmixed land uses that leads to this "I can only go the grocery store once a week" attitude. Life does not have to be that way, people that want it should be able to live like that, and those that don't want it should be able to have walkable lives. Walkable areas mean more accessibility than car-based areas, because stopping does not involve parking.
This is not about "allow[ing] the possibility of not living with a car", all your suggestions increase costs, some increase them a lot. And sure, with unlimited money, I understand that the center of a big city is more luxurious than the suburbs or countryside, this is not controversial. You get to drop the car, which is about 500 dollar per month. But you spend double that on the equivalent apartment (I don't mean equal sized to what you get in the suburbs, I mean the MINIMUM apartment for 5 people)
So yes, costs will need to be comparable, as a basic condition. So using something between a convenience store and a "agent" supermarket is not a serious option, nor is fulfilling every meal by "grabbing something quick", not just from a weight/carrying perspective but from a financial perspective. Making the choice simply based on which is the closest is not a reasonable option either. The problem with a family of 5 is that every cost you have when you are single multiplies by a factor of 3-4, and I'm sure it'll be 5 by the time the kids really get past puberty.
Furthermore, you haven't answered my original question: just simply how to carry it. I assure you a 2-3 day supply of soda for a family of 5 weighs minimum 6kg, and 12kg is better (even that is 3-4 days at best). Carrying this onto public transport, over the sidewalk and/or into an apartment block is supremely uncomfortable. Your suggestions change nothing except making it expensive as well.
Let's take an example: 1 day breakfast + lunchbox + dinner for 2 adults and 3 kids. Take 2 bottles of soda (3kg), 2 cartons of milk (3kg), breakfast cereal (750g), 2 full size breads (750g), something to put on there (700g), vegetables (1.5kg), meat (500-750g), potatoes (2kg), and general supplies (2kg). This is three bags of groceries weighing 15kg. I assure you it definitely won't last 3 days.
How do I comfortably carry this on public transport, and sidewalk, and up a flight of stairs, and you say to carry it on the way home, so in combination with what I carry to work (let's say 5kg) ?
Going once a week, with, say 3 people, carrying something like 20-40kg of groceries, seems vastly more comfortable, attractive, cheaper, easier and faster to me.
This is a common concept in some European cities, with large parking garages directly attached to the subway/train stations.