What about those who would never qualify for state controlled housing or who want some choice or flexibility ?
That doesn't make any sense; if we had huge amounts of council housing, the requirements for qualifying for council housing would be massively reduced or even removed altogether; back in the 60s and 70s when 1/3 of the UK's housing stock was council housing, it was considered a "normal" way to be housed and you did not have to be poor, low-income, disabled, a carer or anything like that.
The requirements have become so stringent in modern times because the supply of social housing is now so limited, meaning it has to be rationed!
In Singapore, for example, about 85% of the popular lives in government housing. I don't think there is any particular requirement, you just apply for government housing if you want to, and you are given it.
I don't think societies HAVE to be centered on social housing (Japan, where I live, is the opposite - we have very little social housing, but housing is affordable because the private sector is so lightly regulated and builds so much). But it's perfectly feasible for the population of a society to have mostly government housing if its voters are happy with this.