That just doesn't tally with what's going on in the TNG era though. It's set less than a century after TOS and the films, and the Romulans have been reclusive for virtually the entire time. There is a big question mark about what occurred with the Tomed Incident and the Treaty of Algeron, the result of which seemed to be the Federation disavowed any development of cloaking technology, and in return the Romulans would leave them alone.
But at the same time, there seems to be a series of conflicts with the Klingons, including Narendra III and Khitomer. Starfleet was aware of these events, so the Romulans weren't completely reclusive.
That map in Balance of Terror clearly depicts the "Romulan Star Empire", which is an area full of stars. It's exactly what it appears to be, a large, galactic power, potent enough to be a very real threat to the Federation should war break out. It depicts the capital star system of Romulus, and their second most important system in the vicinity, Romii. It's like a map of the Federation with Earth and Vulcan marked. The neutral zone was always depicted as a buffer between the Romulans and the Federation along their shared border.
As for the "neutral zone" in the Kobayashi Maru simulation, it doesn't seem to be a buffer zone, because it's full of Klingon ships ready to pounce on any Federation intruder. It's clearly established by treaty though, which suggests to me it's an area of space that Starfleet has agreed not to enter with their Starships. It's "neutral" in the sense that no armed Starfleet ships are allowed in it. If they do, the Klingons are well within their rights to blow them out of the stars. Maybe there's a similar area that the Klingons aren't allowed in? Archanis perhaps? Disputed worlds that the other side has agreed not to militarise, but is open to trade from both worlds. That would be more in the spirit of the Organian Treaty, such as Sherman's Planet. The "Federation Neutral Zone" is referred to by Kruge I think, so it's possible that the Mutara Sector is in this area. Then again, in TUC it sounds just like a buffer zone again, though it's possible it changed in the years between TWOK and TUC.