When someone is born who is a net-negative contributor to the world... it was their parents' doing. They carry their parents' genes; it's a very appropriate punishment for their parents' misdeed to let the child die. It comes very close to being a direct reversal of the original mistake, in fact.

It does sometimes happen that someone otherwise capable of being productive is accidentally stripped of their resources, and ideally they should get some help to get back on their feet - this seems like an ideal use case for a loan. In general, someone will have to make the call that they're worth saving, and I do grant that some people in dire straits are worth saving.

In your example of the old man, it appears to me that he was cheated earlier in life; you postulate that he actually produced a very great benefit to others, and it seems to me that he deserves to have a very great amount of money to show for it. Without government support, he might still have friends to fall back on... if not, then this is clearly a case where welfare does some good, but it doesn't come close to reversing the injustice here. I see the benefit of welfare in this case as mostly accidental, and would prefer that something more targeted be done to repay him, while recognizing his actual contribution.

I just took a brief look at current U.S. welfare law, and it looks like there are some provisions in there to exclude the most obvious cases of people who don't deserve support (able-bodied people who aren't even trying to be productive).