Hi Marmar and welcome to TP.
I am sorry to hear your father has Alzheimer's disease.
I hope you will take advantage of any and all support that is available to you and your father, to make sure he gets all the benefits to which he is entitled and the appropriate level of medical care. Others here can better advise, as I'm not in the UK, but there's attendance allowance, carers allowance, council tax reductions or exemptions, day care, respite care, carers cafes, programs such as Singing for the Brain, the (In)Continence Nurse/clinic, OT support, and so forth. It seems what services are available depend to a great extent on where you live.
Also if you don't already have PoA sorted out, please do this immediately. You are going to need it and it's much harder if you wait.
My golden rule of dementia caring is to always accept all offers of help, immediately. No matter what kind of caring support you provide, but especially if you are a hands-on, live-in, 24/7 carer, you cannot do it all by yourself forever.
Also, if you are the only live-in 24/7 carer, what happens when you get the flu, or break your ankle, or need to go out of town? You need a respite/emergency plan in place.
There are many stories here on TP of people who have kept their PWD (person with dementia) at home for months or years, sometimes a parent, sometimes a spouse or sibling. In other cases, this is not how it works out.
Before you move in with your father, or he moves in with you, you might want to consider the physical layout of the house, to make sure it will be safe (above all) and practical for someone with dementia.
There is a lot of very good information on the Alzheimer's UK website about caring for someone with dementia. For example:
https://www.alzheimers.org.uk/caring
I think the most important thing any of us can do, is to educate ourselves about dementia.
While I'm sure you meant no personal insult to those of us who do not care for our mothers or fathers or spouses or siblings or PWDs at home, and I am not going to justify myself to you or anyone else, there are reasons why residential care (care homes, nursing homes, etc) are chosen.
You may have the opinion that a person with dementia is better at home and in familiar surroundings, because you think they will deteriorate faster out of it. It is true that every person with dementia is a unique case and an individual. However, many of us have PWDs who are NOT better off at home, often for reasons of safety, resistance to care, distance from family, other medical issues that need to be managed, et cetera. In some cases, it's not possible.
Also, the "deterioration" that is often observed when a PWD moves to a care home, is usually down to the dementia, not the care home. Often it is only when the PWD is taken out of their familiar environment, that it is possible to objectively assess their level of function.
As I said, I won't justify myself to you or anyone else, nor would I presume to speak for the many carers here on TP, but before you are so quick to question why those of us with PWDs in residential care aren't caring for them at home (and I recognise that your question may have been rhetorical and may also have been expressing your understandable emotions regarding your father and his dementia), walk a mile in our shoes first, please.
And I would also direct you to have a read of Ann Mac's entire "So Bizarre" thread, which chronicles a family's exceptional attempt to keep a parent with dementia at home as long as possible, or Hilly Billy's "Becoming my Mother's Carer" thread, or any of the many similar ones.
Nothing about dementia is ever simple and easy, and I am sorry you find yourself in this situation. None of us want to be here. Best wishes to you and your family.