Taxi infrastructure is entirely plausible in a project with heavy restriction for private cars.

I.e. build all the roads you want but don't make long-term parkings available.

Fair point, I was thinking about "cargo traffic only" as in the original though experiment.

Coaches with a dedicated luggage section would quite possibly help as well, and it occurs to me that you could have a train with baggage cars ... but having never had to travel with more than one small child I can't say how acceptable those options would be to parents in general.