I will admit I have skimmed the thread rather than reading every post. But I have close, although not direct, experience of this.
MIL cared for her DM at home for many years - which also included taking on her own disabled DBro. Her own MIL did assist a lot with this (retired district nurse), but it was expected at the time that she would just take it on. GMIL died when DH was in his early 20s, but his uncle still lives in the granny annex and is a drain on PIL at times, but they manage (UncleIL is a double leg amputee but has prosthetics and is still, just, getting around on those with walking sticks but will be in a wheelchair shortly, he also has some mild intellectual disability - but will celebrate his 70th birthday this summer and graduated with his degree from Uni last winter).
My maternal GD had a series of strokes which started 10 years before he died, and my GM kept him at home (apart from a couple of week's respite care per year and the odd stint of necessary hospitalisation) for all except the last 6 weeks or so of his life. She was 78 when he died. But she was able to manage most of what was needed herself, and she had supports locally.
She herself stayed at home, mostly alone, until very late in 2011. I lived with her for a year when I moved to this city (it suited us both and she knew it wouldn't be permenant - that was always her worry) before our own house was built and I got married. But DH and I kept very close contact with her since, doing practical things like the garden and a proportion of the DIY, getting shopping in when we visited. And plenty of regular company. She was great, relatively physically fit and able, right into her mid-90s. But she then got some memory problems, and eventually mobility problems, that meant she needed daily visits (which local health services were able to provide) and then increasing levels of these visits (when she'd forget she'd turned on the gas and burn through another pot, for example) - and then with her reduced mobility, she went into hospital for treatment of her legs and then it was agreed she'd move to the long care centre attached when she couldn't go home. None of her 3 DCs was in a position to care for her at that point (between work and family committments, and none of them living physically that close, nor their own houses being suitable).
With my paternal GPs, both were physically fit (for their age) and healthy until their 90s also. Well, GM had ongoing health issues since her 60s, but well managed. And they were also lucky that 1 DD had never married so still lived at home, while 4 other DCs lived within a few miles (with their own families in 3 cases) and only 1 far away. So plenty of support there. My GD was at home until he had the first of a series of strokes, which hospitalised him for the 10 weeks until he died. But up to then, he had been GMs daily carer and companion, with a teacher DC coming in after school a few afternoons a week and live-in DC home mornings and nights mostly.
GM started to get physically frail at that stage, and slowly lost her short term memory. But 1 DC took early retirement, and then the live in DC also, and they still care for her at home. 1 DC is now overseas, but home regularly (and stays in family home). The others all do their bit in different ways - some the physical care, others being the shoulder to cry on or the baker of buns for GM to eat. And they are also incredibly lucky to have the resources to get the necessary equipment (from stannah stairlift a few years back, to renting wheelchair, to having hospital bed for GM now), and also to have some private carers coming in a few hours a week to give them all some physical help (so showers etc are done those times out of preference) and time to get out of the house too. They know how systems work in terms of getting medical care needed, and have brought the hospice on board too, so she can be supported in staying at home in comfort. But this is a large, caring and supportive family who are also in physical proximity to each other.
One of my work colleagues has it in a different way. Their DF is terminally ill at his own house, with their DM having serious health issues. WC has gone PT at work on a temp basis to look after them, as wc lives in the same town and can do it. WC also has all the support services involved and is also able to make their voice heard to get the supports, but only public supports. WC only has 1 sibling, who lives about 90 minutes away, and who is refusing to face up to the diagnosis or their parents needs - so it all falls on WC to arrange diaries and appointments, do a lot of physical caring, and be the shoulder to cry on While their own spouse had lifethreatening illness to deal with last year too (sudden acute issue but ongoing underlying aspect to be managed was discovered). WC is doing all they can to support parent's wishes to stay at home, and can generally accomodate DF until the latter stages (when hospice may be needed), but is having to consider longer term support for DM at the moment and that sheltered living or similar may be needed sooner than otherwise wanted - but WC also cannot continue to only work PT as has own family to support (including 2 DCs).
It is a matter of juggling what is best at any one time.
And as far as I can see, while it is admirable if a family can care for parents in either parent's or a DCs' home, it is not always possible. Either because of the physical/mental realities of the parent's condition, or because of the reality of modern living on the DC and their family (the need to be at work etc).
And, for example, while I did all I could for my GM at home while she was still at home, it was a lot of added stress on me (without much recognition from my DM). And visits to her at home were always tricky and took time. Whereas when she was in hospital, I was able to slip in for a few minutes at visiting time (boss let me take late lunch for early afternoon visiting) and head off again - but much more frequently. And there was a lot less in terms of expectations (no DM or DAunt issuing instructions on checking her meds etc) and physical needs (I had NO problem helping her get places or do things if needed - but there was still a lot of independence to conquer and everything took a long time as her mobility dropped) - whereas in hospital setting, I could just visit and be with her to enjoy her company and give her my time to sit and listen. Rather than NOT visiting during the day in case I couldn't get back to work on time, and then not being able to visit at nights due to DH away and needing to be home with DD.
I certainly don't see having someone in an institutional setting as abandoning them - you still have to look out for their welfare there, but there are people who are trained in the physical and medical care looking after that aspect, leaving you to go back to being the relative rather than the carer. And still giving your relative as much of you as possible.