You're right, they'll probably find a way to survive if he dies, but then it's quite difficult to a lot of providers or breadwinners to think rationally. Because they're used to providing that think that life might just end if they stop providing.
However I think in life even people who looks to be indispensable are actually replaceable, it might be tough or difficult at first, but you're right when you say people need to actually stay alive to do their duties of providing.
RE: If you work too hard, you should check your health