I have heard of this sort of arrangement. I know of a case where an elderly gent (now living alone after the passing of his wife) has taken in a student at much reduced rent in a sort of symbiotic relationship where he stays in his own house with the benefits of companionship and student pays less for accommodation in return for some services and presence. I imagine that it could work well if the personal relationships are good. I also know of a case where a much younger guy lost his wife to cancer at a young age and brought in a live-in au pair to bring up small children (some parallels with dementia care here - small children grow up, patients with dementia grow down)
One of the best documented cases of live-in care must be John Suchet's account in MY BONNIE, where he eventually gave up the struggle to care for his wife entirely by himself and recruited a live-in carer. That is often suggested, but it seems to me that the main carer is then looking after two people, viz the affected relative and also the live-in carer. And the main carer is far from free - respite is restricted although the burden is shared. One of the problems of live-in care is the cost. Care is £20/hour in round terms, and overnight care is typically over £100, so it's not a cheap solution. And it's a very intense relationship. I have seen it at close quarters caring for an MS sufferer and there had to be a regular turn-over of carers because of the emotional and social load. I'm sure it can work, but it's not a simple solution.