Cars aren't necessary, of course; we could get by with horses and carts. But removing them would not "vastly improve" quality of life.
I live in Greater London, about 12 miles from where I work (35 minutes by scooter, 55 minutes by public transport if you time it perfectly, average 1h20 if you don't). My preferred supermarket is 3.9 miles away by road, about 40 minutes by bus or 12 minutes by car (though I normally go by scooter). Would my life be better without a motorised vehicle? My opinion is it definitely would not!
Either I'd need to severely reduce my choices in purchasing or commit to spending significantly more of my life on and waiting for public transport. Neither of these would vastly improve my quality of life.
Largely because public transport is underfunded in London and, of course, ridiculously hampered by the car traffic.
Try getting a bus down the A2 in peak hours or at the weekend. It's often quicker to walk. That's why people end up getting cars and ... making things worse for everyone.
From direct experience: food delivery works well enough for branded, processed and other long shelf life products. It's not bad for meat, as long as you're ordering from a specialist and not a supermarket (and of course there is a price premium). It's not good at all for fruit or vegetables, with current suppliers. Goods are bruised, mouldy, unripe, overripe, etc. It's also useless for bread; you always get yesterday's bread.
I know of families in ecotowns that don't have cars. The tram service covers nearly everything they need, and when they do actually need a car, they get a car-share/rental.
The infrastructure where they live is set up for this, to the extent that owning a car is awkward, because you're not allowed to park it anywhere near the houses, but it's fine because there are such good facilities for people without cars in that area.
The point is that we need cars because we don't have good facilities for not having cars, and we don't have good facilities because we all have cars.
I still use public transport as much as I can, and the traffic in London is such that it's often a viable option, but a car is also a necessity.