I'm having some trouble reconciling this with the rest of your comment. You go on to describe an output (giving money away) that is useful but provides no revenue.
I would agree that a useless product will very often have a hard time generating revenue, but I would not say that a lack of revenue is a strong signal that an organization is not producing useful output. I mean, the software industry is filled with examples of projects that are wildly useful to everyone but are perpetually underfunded and produce very little money for the people working on them.
I would definitely agree with the truism that "if you have no revenue then you don't have a business", but it's just not clear to me where the "because the output is not useful" addition to that truism is coming from.
To be fair, we do ask that: if you have no revenue, then you don't have a business, because the output is not useful. An output can be both useful and not sustainable. Giving away $100 to everyone that high fives you is not sustainable, for example, even though it's very useful to the recipient. It's certainly not profitable.