I agree with you, DR TCB, about ASSASSINS being considered a new musical. It played two months at the limited seating-subscription selling Playwrights Horizon, never had an open ended run on OR off Broadway, so I think it definitely should be considered a new Broadway musical.
The Tonys are a Broadway award. It shouldn't matter how many other places a show has playedaround the country or around the world. When a musical plays New York for the first time in a venue where the run is open and tickets are available to the general public, that's its debut. Any other New York production could then be considered a revival, but this production never was offered to the public at large nor ran long enough to transfer to a theater where it could have had an open ended run. Therefore, I think it should be considered a new musical to Broadway and have its score considered as a new score.
However, I'm betting now that it won't be, and the score won't be eligible for Best Score either.