It is not just that. The mere act of shopping at most stores requires a car.
In Japan for example all fairly big items (such as chairs, etc...) gets delivered. You do not buy 24 cans of coke and a liter of mayonnaise and take it on the train with you. You buy one ridiculously small can of coke at a small (and expensive) convenience store near you.
So, I wouldn't be as quick as to require all people shopping to not come with a car.
Living in the suburbs has its advantages - cheap storage space, etc... So you can actually buy in bulk.
No one is advocating restricting parking. If a store wishes to provide underground parking, for instance, they should be permitted to do so.
Interesting enough, I've seen the opposite effect to the one you've described - in a city where all street parking was costly, a supermarket had to make his parking paying for non-customers and customers who took more that one hour.