"Most people without dementia are at home with carers popping in. If someone needs getting out of bed, washed, dressed, food given and toileting, then carers visiting their home is usually what will be recommended."
Sure. 
And when they get to the next stage, when without dementia, they are so confused, frail and unsteady on their legs that they are not safe to be left alone for even short periods?
Who does their shopping? Who does the laundry? Who makes sure they don't run out of incontinence pads? Who pays the bills? Who gets endless phone calls from the neighbours?
How about when they try to use the microwave even when they've been told dozens of times not to, forget how long something needs to cook for and nearly set the house on fire, and you turn up to a house filled with smoke?
When the carer 'pops in' and give them their tea and goes again, and you turn up an hour later to put them to bed and find them on the floor in the dining room because they fell asleep in the middle of their dinner, and keeled over sideways off the chair?
When you have to wait a very long time for an ambulance?
When, during their time in hospital after that fall, they are told over and over again to ask for help in going to the toilet, yet forget to ask and try to get up, falling flat on the floor again?
When you get a phone call at 1.30am from them saying that the carers haven't arrived to give them their breakfast yet, and you have to explain that it is the middle of the night, not the morning, and they need to go back to bed?
When they forget to hang the phone up properly all the bloody time so you can't contact them at all and have to drive 10 miles over there at whatever time of day or night, just in case there has been another disaster?
When you can never, ever, have a drink in case you have an emergency to deal with and have to drive over there?
When you have to spend hours and hours every day speaking to care teams, support people, nurses, doctors, never minding that you are losing money every day because you can't go into work?
Because all of this (and a lot more) has happened to us in the last three weeks, and it is getting impossible. My DH (no spring chicken himself and with a heart condition) is having to move in with her now for a few days, because the hospital in its wisdom has discharged her again today, with no time to rearrange for carers to come in at such short notice.
If you had to deal with that and the sheer exhausting unrelenting stress that goes with it, perhaps you too might think that maybe a nursing home isn't such a bad idea after all.
But hey ho, she insists she wants to be at home, and there we are.