I'd guess that in common speech most in the US would consider the "the" redundant, as there is only one Yale university. The "the" might be used in very formal written matter but would not be common in speech.

An acquaintance who did some typing for an admisssions office administrator at MIT (what most people called it) about 50 years ago remarked that he insisted that it always be "the M.I.T.," but most others in the office left off the "the." The particular administrator was otherwise considered a bit peculiar, as he was British, and my acquaintance was forced to find other employment as his accent was unintelligible (to her) on dictaphone recordings.

A "confounding" factor in that case is that most others would have considered it appropriate, when spelling it out, to say "the Massachusetts Institute of Technogy" but would use "M.I.T." or "MIT" as an acronym (without "the") rather than as an abbreviation where, for technical "correctness" it would be appropriate in formal usage.

It should be remembered though that the distance from New York on the east coast to Sacramento on the west coast is about 2/3 as far as from New York to London, and there's lots more "culture" in the intervening space (unless you consider fish as social animals); so there are lots of regionally deviant speech practices in the US. Generalizations about what "an American" would say are likely to be fairly questionable in the next county, much less at opposite ends of the country.

John