Dementia can often be cared for at home. It usually works best if the person with dementia lives alone. As a PP said, the main reason people end up in care is due to carer (informal carer e.g. relative) breakdown. The vast majority of older adults are cared for in the home, until they die. This the preferred option for those funded by the LA due to it being cheaper.
I would never ever recommend a relatives moves someone with dementia in to their home, for many reasons, but the change of location can cause a seeming deterioration of the dementia symptoms due to loss of familiarity with their surroundings. It is also not something you can simply just "do" unless the person with dementia has capacity to make the decision for themselves or you have LPA. Often times, if they do not have capacity to consent to the move, a Best Interest meeting is needed and the Social Worker or Mental Health specialist will make the decision and as I said above, we don't like moving people in with family.
There is a strict criteria for those entering care homes under LA funding, the main ones being night needs and inability to maintain safety (e.g. those who try to leave their property but have lost road safety or sense of direction and frequently get lost, or those who fall frequently and refuse to remain seated.
They don't all need nursing care - some need simply "residential care". The difference between residnetial and nursing care, is that n nursing homes they have at least one registered nurse on each shift in addition to trained carers. The NHS pays for the nursing element of care (around £130 per week).
REGARDING FUNDING:
Currently those with less than £14500 in savings or assets have their care funded by the LA (whether at home in residential care). The problem is that each LA has a set amount they will fund per week, and often this bears no relation to the cost care homes in that area charge. Since the vast majority of care homes are now private, the LA have no say over how much they charge. This results in people having to move area which can then result in loss of visitors and links with family. Even someone with savings, who needs some support with fees from the LA wouldn't be able to go in to a care home that costs more than the LA rate as they would require a '3rd party top up agreement' to be signed, which cannot be signed by the person themselves.
Care homes are expensive, but care homes are not massively profitable, BUPA recently sold most of their care homes due to the profit margin not being enough.