You got me on language at least, but it's not that simple. It was overturned for specific "private" spaces-- e.g. the Oval Office-- but the most recent Court of Appeals panel from June 6th did find that the East Room is not private like that and kept in place the prior ruling that the AP's allowed in the East Room. (The NYT article you linked doesn't mention this, but it's really only talking about a July 22nd denial of the AP's emergency petition challenging the partial overturning, not any change to the situation on the ground.)
Regarding the East Room, according to the White House Correspondents' Association website (https://whca.press/covering-the-white-house/), reporters either need to have a hard pass or a temporary pass from the White House Press Office to enter. The Press Office generally only gives press passes to the press. That's why even if we restrict the conversation to the East Room, it's still a "special right" (or special privilege).
Regarding the Oval Office et al. (which admittedly is what the term "press pool" actually refers to), the Trump administration's entire argument was that press access to those spaces is a "privilege, not a right." So keeping in mind the courts affirmed an actual right to the East Room, the White House itself still grants other journalists a special privilege (because they're journalists).
You can make an argument if you'd like that the guy earlier in the thread should've said, "The press has special privileges because accountability and informing the public was supposed to be their job." (This ignores that it's still a "right" in other cases, like the East Room.) Squabbling over that word choice doesn't do a lot for the topic of this thread, which was journalists no longer serving the interest that justified giving them those privileges/rights.