Actually, I know exactly what's going to happen (to me) in the future: I'll eventually grow old, get sick and die in pain. Or maybe I'll just get the "die in pain" part. That's how it goes, right?
And in the past I was very young and couldn't yet fend for myself. I benefitted from being born in a developed nation and, like you say, if my parents had died while I was young, I'd have been taken care of (though in my country's case it would probably be my extended family that'd have looked after me, first, and the state would only intervene if I had noone to care for me).
Like, I totally get what you're saying, but a significant number of people don't see it that way. Maybe they forget they were once too young, or that they will one day be too old. Maybe they just think they can do well enough for themselves that they'll just wing it.
My point is that for many people contributing to things like public healthcare doesn't make sense.
For instance- in the UK education is not free, and you will hear lots of people saying, indeed "why should I pay for someone else's education?". That's much more common than grumbling about having to pay for the NHS, but in the end it's just the same thing.