I am closer to being your dad than some others posting here, OP. You may be interested in things from the other side, so to speak. My children and grandchildren live relatively far off (some abroad, indeed), and I am gradually becoming less able to cope with the vicissitudes of age and all that. I live in England.
What do I want from my children? My top priority is that I do not want them giving up their lives to look after me. They have their own lives to live, their own children to care for. And I love them all dearly. They do me, too, which means they sometimes need persuading I do not need them to care for specific aspects of my life.
I have a decent pension, some savings and a house worth a small fortune in a beautiful location. So I get what help I can from social services (minimal given my financial status) and pay for help from my income and savings. When these latter run out, or if needs must, I will move into a care home and eventually sell my house to finance ongoing care.
It would be nice if, as a society, we could have worked out sufficient communal provision to see everyone well looked-after in old age, but, sadly, too many of my fellow citizens are too selfish or ignorant for this to have been managed so far. So these my own arrangements are faute de mieux.
If I die before my savings run out, my offspring will inherit the remnants as well as the house. If not ... well they may inherit next-to-nothing in the end. That will not be too bad, I think -- they all have decent jobs and so on, partly as a result of their upbringing ... and as my youngest said when we were discussing all this, 'We have already had a great inheritance from you, Dad; we do not need any more money when you die.' I agree with that.
Meantime, I hope to see children and grandchildren regularly -- but at their convenience (see 'they have lives to live ... ' above). And so, I will continue to enjoy my developing dotage.
Oh, and I have an Advance Directive (ex- Living Will ) saying DNAR etc., if I become too disabled. And I plan to 'exit' (via my 'Switzerland Fund' , currently) if it looks like I am going to get too gaga; I want my children and grandchildren to remember me as a more-or-less fully-functioning human being. I hope this all works; at least I have plans in place, and whilst I know the best-laid plans ... etc., I have done what I can and can do no more.
I recommend all this as a recipe for happiness in old age.
[Btw I second pp's advice re 'equity release' schemes: avoid if you can, they are almost universally run by thieves and scoundrels.]