No single mode of transportation solves every use-case. My car is suboptimal when I'm going downtown, as well as for trips longer than 300 miles or so.
At an ordinary population density, supermarkets are about 2 to 5 miles from each other, which is probably 3 to 7 miles by road.
Large stores require many customers in order to remain in business. If supermarkets are closer, then they lose efficiency. Prices must be higher. Variety drops; you'll get just 2 types of cheese and 1 type of apple if the stores have to be smaller.
Without a car, large shopping trips become impossible. A young healthy adult can carry 1 or 2 shopping bags about 0.7 mile without major difficulty, but that just isn't enough. Stores would have to shrink by a factor of 20 to 100 in order to get them close enough, but the efficiency still plunges. Think of the portion of each sale that must support the manager's salary, or the number of stores at which each delivery truck must stop.
Without a car, frozen food can only be purchased in winter. Hot food, such as the fried chicken my supermarket offers, would never be able to get home in good condition.
I did the maths a while ago; even if every house has a quarter-acre block (subtract actual street space from that to get a slightly smaller block) you can fit 4000 houses, about twelve thousand people, within a 1km radius of a mini-downtown point. Twelve thousand people is certainly enough to support a supermarket, a bunch of restaurants, cafes, specialty stores etc (think of what a town of twelve thousand people looks like). The only trick is actually zoning rows of shops in appropriate places.
Of course I still drive to the supermarket anyway, unless I’m only buying a couple of items. This isn’t an argument against cars, it’s just an argument that low(ish) density housing and walkability aren’t incompatible.
More broadly I think that Americans looking for solutions to their city-planning problems should look less at Europe (or god forbid, Asia) and more at Australia, since I think we’ve done a decent (though as always imperfect in implementation) job of combining European and American city planning virtues.
Couple days ago I literally counted >100 varieties of marinara spaghetti sauce in the store. There's no way a walking-distance grocery is going to carry anywhere close to that (ripple effect on to all the people employed in making each specialty) AND do so at Walmart prices.
I live close to the grocery, and there's still no way I'm going to carry supplies for 4 that far (recall as a child doing similar with my mother, that was quite a trek made possible in part by not having a job).
Y'all who want pervasive public transit and no cars, yes I see your point - and really don't want to live that way. Do as you like, just don't do it in a way that compels non-urban living to comply. 'scuze me now, I have to get the kids from school - driving them because herds of children on mass transit is a haven for bullies.