“nobody will work if they don’t earn money from it”
No, it’s just a tautology resulting from how work is defined. Anything you do on your own time (not paid) is by definition not work. It’s a hobby, a chore, volunteering, etc. By contrast, anything you’re paid to do but would stop doing if you stopped being paid is work.
Looking after a sick relative may feel a whole lot like work. The difference is that if you don’t do it, who will? Asking a stranger to care for your relative for free just isn’t done. That’s why it’s work if you’re a hired nurse but not work if it’s your relative and you do it for free.
You can do this exercise with anything. Go to a restaurant and you expect to pay for your food after you eat. Go to eat dinner at your mother-in-law’s place and you don’t expect to pay. In fact, she’d likely be baffled at best and insulted at worst if you tried to pay her for your meal. Cooking food for strangers is work, cooking food for friends and family is not.
Anyway, don’t you think there’s some validity to the original statement? Why would you expect strangers to cook for you and care for your sick relatives for free? Note that even if you’re not paying them but someone else (possibly the state) then it’s still work.