You can definitely do statistics on ordinal data, but it is a lot less fun and believable. If used in something politically sensitive, it will be controversial. We use it in medicine, and we only suffer from profit motive, and I don't like them very much. But we have the tools.
My point would be that if this is a purely theoretical discussion, it’s fine to keep things as an abstraction, but when it comes to implementing some sort of public policy I imagine there needs to be some kind of metric used in order to function; welfare disbursement, for example, being based on some measure of income and not just “well, they look poor enough.”