Spot on. Council tax will probably go up as more tenants who are currently working (it won't be 100% but a sizeable number) decide to claim housing benefit as the rent doubles. And since it will be deemed as "fair rent" the council will be obligated to pay all of that on housing benefit.
Your council tax will not go down, rents in both sectors will go up, and property building companies will start their own social housing business.
I suppose that if employers of shop staff, bus drivers, street cleaners etc suddenly decide to pay their staff more, this won't happen. But that's very unlikely - they'll just continue to pay the same to EEs and the local working population will decide why work, when they can get the rent paid with HB? I realise this already happens, but double the rent and surely it will get worse?
The logical conclusion of all this (and which really would reduce house prices) is to do away with housing benefit altogether. But I can't see any current government taking this option - it would make last year's riots look like a vicars' tea-party. You can stop hb being paid to private landlords charging the earth, but stopping hb on "fair rents"? Be realistic, it isn't going to happen.