I think, Vybs, that you do not fully understand
matrimonial home rights.
It applies in a situation where the marital home is in the sole name of one spouse.
In such a situation, the non-owning spouse doesn't have the right to live in the marital home by virtue of her ownership ( because she isn't the owner or even a joint owner ) but because husbands and wives have a duty to support each other and keeping a roof over your wife's head is a very important part of that duty.
Matrimonial home rights apply only to the marital home and not, for example, to the family holiday cottage in Sunnycliff-on-Sea.
The point is that, where the marital home is in the sole name of one of the spouses, that spouse can sell or mortgage the house without the other being aware of it.
So, if the marriage is heading for the rocks, it is prudent for the non-owning spouse to register matrimonial home rights to prevent this.
You can't register matrimonial home rights against property which is rented.
As regards the Housing Department, my long experience in local authorities has convinced me that some of them either don't know what they are talking about, or they do know, but they rely on the ordinary person's lack of knowledge and bamboozle them into
believing that they have no rights, when they have.
I very much sympathise with Housing authorities who are starved of resources by a Government that puts duties on them and then denies them the funds to carry out those duties.
What they are telling you seems tor me a load of eyewash, not to put too fine a point on it.
My advice is to find your local branch of Shelter. If someone has dependent children, (s)he is likely to have a priority need for re-housing. Make an appointment with them URGENTLY.
LMM