Or you end up with fist fights and road rage over limited resources. And it's very easy to say "limit access to cars", but it absolutely requires a build-up of public transit to make up for it, which usually goes beyond local zoning practice
The High Cost of Free Parking (https://en.wikipedia.org/wiki/The_High_Cost_of_Free_Parking) is pretty eye-opening in terms of how non-drivers end up bearing the cost of parking minimums.
Car ownership is lower among people with very low incomes, but if they want a place to live they still have to pay for parking to be built. Shoup’s book has some examples of proposed low-income housing that ended up getting blocked over parking minimums, even though most of the would-be residents did not own cars.
Hint: no it won't, because markets don't work in delivering large-scale infrastructure for the commons