There are a number of different schemes whereby a local authority gets involved in renting a house, and not every rental under these schemes is the same. Different councils have different contracts and even any given council changes their contract over time.
Are you looking at the RAS scheme, the HAP scheme, or something else.
I have a house let under HAP. And I am very happy with it. Rent paid into my account every month on time.
I choose the tenant myself, I think that is important, as if there was a bad tenant, the council would see that as your problem rather than theirs. My tenant had been renting from me for more than a year and we had a good relationship before I suggested that we apply for HAP.
If you let the council select the tenant, they may be glad of an opportunity to house a difficult case.