Well, there might actually be an arbitrary "Now you can unlock NG+" point. I just did a pre-run using cheats to see if it was possible to unlock NG+ upon entering the EoT on a first run, and apparently it is not. When I typed up my initial question, the "under any circumstance" was actually referring to your three options after the Blackbird, although the fact that I hadn't even been thinking of endings other than "Beyond Time" and "Reunion" made me totally overlook what I was actually saying. *attempts to remove foot from mouth*
To (attempt to) answer your question, though: since there does seem to be at least some threshold you have to pass before NG+ will unlock, perhaps it is when you open the door to the ocean palace, since idioticidioms has stated that he has, indeed, unlocked NG+ by defeating Lavos at the Ocean Palace. What I know from personal experience, however, is that I was able to unlock NG+ the first time I ever beat the game by defeating Lavos via the bucket, without venturing through the Ocean Palace. Perhaps the deciding factor is getting the "Beyond Time" or "Reunion" endings (or, perhaps, "The Dream Project," if what idioticidioms stated was true).