Saying you went to "a Ohio state university" in a spoken sentence is quite ambiguous; you could have gone to Toledo, or perhaps Cincinnati, or Miami... it's unclear. And unlike the SUNYs, they aren't all part of the same organization (or is SUNY actually part of the same organization?).
Saying you went to "a OSU" is even worse, as then no one knows if you went to Oregon State, Oklahoma State, or Ohio State. People probably won't guess an Ohio school other than The Ohio State University with the abbreviation, but they'll look at you funny and depending on where you are, might guess one of the other O states.
Also, in regular life, it's common to refer to it as "Ohio State" or "OSU" (assuming Oklahoma/Oregon isn't ambigiuos in the context). You'd probably think that the Queen's initial tite, "Her Majesty Elizabeth the Second, by the Grace of God, of Great Britain, Ireland and the British Dominions beyond the Seas Queen, Defender of the Faith" was pretentious, too, but I don't think the point was to sound modest. The "The" does attract plenty of conversation... which isn't necessarily a bad thing.