If it was a long crawl through a traffic jam every time I went to get groceries I'd be tempted to get a good sized garage ready freezer [1] and a larger car so I would only need to drive to the grocery store once or twice a month. There's a small grocery store within walking distance that is too expensive and doesn't have enough variety to be my main grocery store but would work for those things that won't keep for a month.
[1] A normal home freezer that is part of a fridge/freezer is typically is designed to work in a room around 55-110℉ (13-43℃) and a normal stand alone home freezer typically works down to 32℉ (0℃). A "garage ready" freezer is one designed to work when the room temperature is 0-110℉ (-18-43℃) so if you don't have room in the house you can put it in your garage.
Since the beginning of the pandemic most supermarkets have a decent delivery service. I pay 1£ for delivery (4h window) and when I was in the EU, my store of choice gave delivery for free if you purchased some specific items.
I think my ideal utility vehicle would be an electric van...
Grocery delivery in my area is expensive ($10 or so) and they usually don't give you everything you ordered. My annual cost to own a car is under $1000 including gas, so it doesn't save much money to ditch the car.
For many people moving stuff from shops to their home is their primary use case for an automobile. Why wouldnt they buy a car tailored for that purpose?
If you have a sunken investment, then it makes sense to use your already-depreciated car for grocery... But otherwise, assuming you have access to affordable delivery service, you might as well save yourself the effort and cost, and only use a car for the other needs in your life.
YMMV of course, you could go for a smaller and cheaper to run car, but if you have a family you'd probably need 4 comfortable seats...
Plus Costco doesn't deliver, and that's where many of us do our bulk shopping with large vehicles. (You can order from Costco through Instacart in some areas but it's one of the worst of the delivery services.)
This is high on the list of BS claims I have seen. If it were true, why wouldn't car companies use it to sell cars?
The truth is you'll probably be driving in traffic or crawling in a traffic jam… of which you are the cause. They'll never, ever show someone struggling to park one in a standard space at an urban supermarket.