Taking a generalist approach, there are many factors why even in recent past many elder people were supported by adult children.
In most countries in the early 20th century there was no retirement insurance plan provided by governments, and people couldn't afford to save money, as well; The first initiatives of retirement pension were started by private companies for their employees, who couldn't work as good as before (as a way to "get rid" of an employee that doesn't give profit); Later on, the people who could, paid banks for a private insurance plan (the people who were poor hardly have the chance to save any money).
In the past, life expectancy was lower and there were many impairing diseases much more prevalent, what impeded people who managed to get to old age to work.
This is some reasons why many elder people couldn't depend on themselves for a living.