yes, post war town planning did regard off-street car parking as desirable and maybe even a human right.
On the face of it, as well as being convenient for residents, it might seem like something that's good for the public realm - get the clutter of parked cars off the street. But parking needs a lot of space, and very often what happened is that the parking became a kind of dead zone between buildings and streets, something that is completely unfriendly to pedestrians. Either, as the windswept open car parking areas that have to be walked across to get to the entrance of buildings (there's a little bit of that on the Loughborough Estate, although not as bad as many) or solutions that involve using up the lower floors of the building (Moorlands) or the later example you give of the ex furniture shop. That also creates dead frontages. There's another example of that along acre Lane, presumably from the same era:
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All these are just demonstrations of how accommodating large numbers of private vehicles is fundamentally at odds with creating environments that are good for humans to live in.
Postwar, modernist architecture has a bad reputation with many people, because of associations with these kinds of unpleasant environments. I often like to argue that the main problem with that era of architecture/town planning is mostly to do with the fact that it attempted to embrace the private car. Most of the bad things result from that (which determined how buildings were set out in relation to streets) rather than from the architectural style of the buildings themselves. At that time, people were naive to what would actually happen. The images are all of sweeping highways and parkland with no congestion or pollution in sight.
And still in the 80s or 90s, we were building these stupid buildings like the ones above, with car parking built into the ground floor to pretend it didn't exist.
Plus the kind of suburban-style closes that exist along Shakespeare Rd.
None of these approaches result in a good solution because the fundamental problem is: too many cars. These approaches encourage car use
and they let it dominate the urban environment.
And it only seems to have been in the last 20 years or so that this has finally been accepted, and it's now quite uncontroversial in somewhere like Lambeth to specify that new devlopments are car free.
Although much of the rest of the UK ploughs on with car-is-king approaches to planning.