True-ish. But try talking to some old folks, about the vastly reduced scale of such volunteer work since (say) the 1950's. That "voluntary unpaid" stuff would include religious organizations, civic groups, housewives, ...
More concrete data are hard to find, AmeriCorps' only started in '02. We can look at volunteer rates across demographics[1] but it's likely true that volunteer hours drop off with age.
[0] see: Bowling Alone (needs a post-COVID update, though)
[1] https://americorps.gov/sites/default/files/document/Voluntee...
It’s true that first you care for yourself but after survival is secured many people care for others without profit concerns.
But if you need more time to satisfy your own needs you have less time for others.