The Equality Commission have pointed out to you that it's not an equality issue. You can't successfully base it on Housing Benefit and ignore the overall stats (which are available at Gov.uk although due to some data being excluded for confidentiality, the totals below don't always add up but I've just copied and pasted!).
To May 2016 - total number of HB claimants - 4,681,714
Single Claimants - 2,527,753 (Male 1,187,802, Female 1,325,542)
Single with Child Dependants - 1,146,461 (Male 57,639, Female 1,078,162)
Couple no Child Dependants - 442,609
Couple with Child Dependants - 564,896
If you look at the tables by age, there is a massive jump after age 65 (for singles) - does make you worry for the future.
I hope you find suitable accommodation for yourself and your family OP, I can sense your determination (and understandable frustration about this issue) because it is a situation that causes a lot of problems for a lot of people. The lack of social housing is a long-standing issue with no quick resolution unfortunately.