>the child of someone else will have to care for them in old age.
The old person might want someone else's child to care for them. But someone else's child will not have to care for them.
The old person might want someone else's child to care for them. But someone else's child will not have to care for them.
Generally it's considered a burden of society to take care of their elderly. Thus 'someone else's child' will have to take care of the childless elderly in some form or another.