This looks like a classic case of "Who gets to wear the crown?" to call themselves New York State. There is a chancellor of the entire SUNY system. Wouldn't they be able to make that determination? I think Cornell is actually New York's land grant college, and NYU has the flagship name. So they are going to argue over who gets to be New York State?
Cornell is indeed the state's land grant institution.
And while the Chancellor can make a recommendation that a particular school become the flagship university, the final say will come from the BOTs which are all appointed by a governor and approved through the state legislature process. So they will be wary of the politics involved with such a move.
Of the current 16 BOT members (minus the student trustee), 8 of them have all of their education from private colleges. three a mixture of private/public, and 5 public only. Of the latter 5, 3 are only SUNY schools. Of the 8 from solely private education backgrounds, 3 of them are from NY private schools only.
The two schools that would have the best chance of being designated as THE flagship university of NY, imho, are UB and SB.
UB, though founded in 1846, was a private college until the early 60s when they were bought out by the SUNY system. SB was founded in 1957 as a SUNY university although then called State University College on Long Island.
There simply is not enough vested into either to make the state as a whole rally around any SUNY school in the same manner as the Michigans, Ohio States, etc.
How are athletics funded at the different SUNY schools? Does Buffalo have the capacity to stand out? Stony Brook would have to make the transition to FBS. Their stadium is still a bit small for that.
And speaking of NYU, why do the Violets not play sports at the highest level? They are certainly big enough, and they aren't an Ivy. They could play their football games a Yankee Stadium.
The SUNY schools are poorly funded by the state in terms of athletic monies. As for your final question, I don't have an answer, but I'm certainly glad they never did.
Cheers,
Neil