This seem to imply that shops don't benefit much from parking when majority of customers lives within 1km of the store, and if that's the definition of local businesses then I don't see much problem eliminating parking. Local businesses catering to local customers. Non-local businesses, ie businesses where the majority of customers isn't with that 1km, should not be located next to shopping streets, which might require a new definition of shopping streets which centralize all kind of shopping to a specific street.
This seem to somewhat match a trend that I have noticed here in Sweden. Stores catering to a larger audience seem to have started to move away from expensive city cores in the last couple of decades, with specific shopping street decreasing in importance as a result.