Nevertheless, the examples you've stated are easily dealt with by allowing as-needed exceptions for the disabled. I don't see what moving houses has to do with wanting to go to the town center in a car though.
I think what he means is that when you want to move a couch/bed/big television for example you don't really have good options, because even disassembled these are usually pretty bulky boxes to move through public transport or by bike.
In Berlin, where I live (in the center in fact), we're big on reusing and recycling. I don't own a car, but quite often I need to transport bulky second hand items from wherever I bought them, or bulky trash to the recycling center. So I drive one from a car-sharing app.
I've also needed to take someone to a hospital, and it was just quicker to grab a car2go than waiting for an ambulance.
My point is, there are many use cases where the ability to drive or call a taxi is better than the existing alternatives. I agree that cities everywhere need far fewer cars in them, and I think Berlin with its focus on biking and public transportation but availability or car-sharing vehicles for on-demand private driving is heading in the right direction without having to outright ban cars.
We SHOULD use cars. But we should treat them as garnish, not the main course. Use them for exactly the things they're great for: moving houses, needing to visit the hospital, long road trips to a destination where you'll need flexibility to drive around once you reach etc.
I'm not advocating a ban on cars. I'm not even advocating not owning them. The simple step of banning them on the main shopping street is a far cry from any of those things, but makes the actual experience of being on that street a lot better for the shoppers.