intentionally made to a public forum could be considered private information after the fact

Well that's the thing. The criterion is Personally identifying information. Not private information.

Remember GDPR includes right to be forgotten. Person is allowed to change their mind. At one point they might have wanted and agreed for that information being readily publicly available. Then they have right to change their mind "Nope, don't want the information out still".

As I said. Just because it has been publicly published, doesn't remove the protection categorization GDPR offers.

It is just then PII you at the moment want to be publicly available. Ofcourse deleting anything completely of the net later is not possible, but the point is when informed of deletion order, that organization is not supposed to be part of the "this persons information is published, when they don't want it" problem anymore. Company can't control all of Internet, but they can control their own conduct and within that limit they must comply to privacy order. Even if it doesn't perfectly swipe the information from all of internet.

It is utterly different mentality and regime from "private/secret" or "public/its gone now" system. In this other system privacy is on going process and scale. It can move two ways instead of just unidirectionally. Person has right to ask and demand for what has been public to be made more private. As they also can choose to make private more public.

EU and its citizens have right to choose what principles they base their privacy laws on and they chose this different kind of regime. Other regions and countries are free to choose otherwise in their own jurisdiction (though EU does this super claim of "EU data subject involved, we claim jurisdiction")