Because countries have specific laws about what constitutes acceptable documents for entry into a country and the usually include things like “issued by a countries recognized and lawful issuer of passports”?
Most immigration decisions are negotiated and often reciprocal (you give me visa free travel, I’ll give you visa free travel).
Unless a document is specifically recognized as issues from a non-state entity (e.g. UN refugee passports) they aren’t valid travel documents anywhere.
Like you say yourself it is possible that non-state entities can have their travel documents recognized in some cases. I don't see any reason why that non-state entity couldn't potentially be a US non-profit. Obviously in this particular case, this agency's documents are not widely recognized, but that has nothing to do with their legal status in the US.
Most immigration decisions are negotiated and often reciprocal (you give me visa free travel, I’ll give you visa free travel).
Unless a document is specifically recognized as issues from a non-state entity (e.g. UN refugee passports) they aren’t valid travel documents anywhere.