Rephrase the second point as "everyone must work for a living", and it becomes clearer. Your second point becomes a corollary: people can only be cared for if they work. Your last paragraph is only possible if we want to allow people to live without working.

But many would see that as morally repugnant. Not everyone, but a majority, and especially a politically powerful plurality. That is what you will need to deal with to make a change.