There is no guarantee your children will visit or take care of you in old age, and if you're having children due to the above reason, that is a very poor reason indeed to have children, as it seems to be more of an argument for the parent's well being, rather than the child's. If you don't want to be alone, cultivate lifelong friends.
These are all novel cultural values not common in any traditional society with normal birth rates. In fact, this attitude is one reason why birth rates are so low.
It is totally normal to expect children to care for you in your old age. The guarantee is that other people can and should ostracize people for not caring for their parents. When an acquaintance tells me they don't visit/call/care for their parents with pride, I make a mental note to not be friends with that person. If you can't keep the most basic relationships straight... that's not a great sign, realistically.
That is certainly...one opinion. You don't know what those people have been through, they could have had abusive parents for example and don't talk to them anymore. If the parents don't take care of the child well when raising them, I consider it more than fair to not take care of them in old age (or even simply once the child leaves the nest, so to speak). Taking care of one's parents is not an immutable law of nature (indeed, many if not most organisms simply breed and leave their children), nor should it be. That you implicitly "make a mental note to not be friends with that person" is quite telling indeed.
As a parent myself I can tell you that they are very likely referring to having two to three children close together rather than a single child ... ie. having both a six year old (eldest) and a child in diapers | stroller (youngest) at the same time.
After that period things are great!
. . . until you've faced with a household full of teenagers rebelling against anything and everything in overly dramatic ways.