Where did you read that the population has been declining Rupert? I doubt that is right. Immigration started in the late 1950's for a start, which would have upped it straight away. Of course people (including myself) have moved out of Birmingham, but those that were left had children and those need houses - I really find that quite hard to believe.
Council houses are not peculiar to Birmingham, they have them in most places, although some of the posher areas may not have any, there are some relatively nice areas where there were a few also - I say were because a lot of tenants purchased theirs. Councils were allowed, under Margaret Thatcher's government, to offer their houses for sale to the current occupier. That took a lot off the market, wrongly in my view. Prices are high, ridiculously so, but to one who was purchasing, at any time, the price has always seemed high.
When manufacturing declined, the people did not move away or die, they were out of a job, but still had to have somewhere to live. That situation has gone on and on.
I think living abroad you can never ever imagine how life has changed. My sister and brother both live in America, my brother rarely comes back (and never speaks to anyone anyway). My sister is always very shocked at the way things have changed, and that is in just 21 years. She comes back about every five years or so.