My grandparents are dead, and never required the support of myself or my parents. They either died while they were still of working age, or died suddenly - and were in good health and required no further support than their own pension prior to that. I hardly imagine that I'm alone in that situation, so your broad assumption that everyone lives to such an age, or that their health deteriorates to such a point that they require extensive or prolonged care is a bit of a leap in itself.
I've no idea what it's like where you're from, but where I'm from my contributions fund my own state pension and general healthcare and social benefits (unemployment, incapacity, housing, so on and so forth). By the time I retire (if I'm a bit lucky), I will have taken very little from that - maybe some unemployment cover if I'm unfortunate and the occasional illness.
Someone who had children, on the other hand, takes from the social pot their maternity benefit, their child benefit, if only one of the parents is working they'll be getting housing benefit, possibly council tax benefit. They're not paying in any more than me individually, but getting a heck of a lot more out of it. They'll still be getting their state pension when they retire too.
Of course I don't begrudge them that, and yes, one day their children will be contributing. But when their kids have kids, the cycle begins anew.
Also, you'll find that anyone over the age of 25 with a lick of sense and a little bit of disposable income is already making sure that they're putting money aside for their retirement to ensure that they aren't a burden on anyone - the state OR their children.
So don't try and tell me that I'll be a burden on the state when I'm old if I choose to not have children, because it's nonsense.