What I'm hearing is even if you do save up for a higher quality nursing home/long-term care facility you still need to hire a third party advocate that knows the patients exact needs and conditions. They also need to be around almost 24/7 and not take vacations because their substitute might not know all the patients specific needs.
I guess this becomes a personal decision on how much one wants to save for future possible health care costs vs doing the things that make them happy now. I think most people approach this situation with the mindset of I'll save what I can for the future but also live a worthwhile and full-fulling life now, so that if something does happen that reduces my quality of life later on or even in the near future then so be it, I have lived a live according to my values and I can die with no regrets, however painful my final moments maybe.
There are just so many things that are out of our control and your perspective from a healthcare perspective is very similar from a financial one. I'm not saying that your concerns are not valid however there comes a point where trying to prepare for every single outcome becomes impractical or cost prohibitive.
Another thing that I can relate with OP is that nursing homes do scare the crap out of me, but not for the same reasons as the OP. I'm not so much concerned with the quality of care, other than the abusive ones where there are videos of caregivers clothes lining patients, I am more so concerned about the lost of autonomy and familiar environment. The book 'Being Mortal' comes close to summarizing how I feel about end of life care.
I have a grandmother that was moved to a nursing home after my grandfather died. The used to live in their house where they raised five children and were retired for 30 years. During the last years of my grandfather's live they had two full-time care givers taking care of them. They were close to family and friends. Even though they did not get the best medical care, they lived a life worth living. No my grandmother is in a nursing home that is far away from her old house, family, and friends. Only her youngest son looks in on her twice a week for an hour or so at a time. For me I would rather risk falling in my own home, close to friends and family, having the autonomy of eating and bathing when I want than to, than being in a unfamiliar place around people I don't know, being told when to do things.
I think people not in the situation, such as the patients children and caregivers, put a higher priority on safety than the patients actual happiness, condemning their loved ones to a prolonged and miserable life.