An old soc.history.what-if post of mine, very slightly modified:
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In January 1899, William Howard Taft received word (from his brother Henry Taft) that he was being considered for the presidency of Yale. He wrote a letter to Henry, declining the job, giving as one of his reasons his religion. Yale's strongest support was from "among those who believe in the creed of the orthodox evangelical churches," and with him as president, Yale might be deprived of that aid, because he did not share that creed:
"I am a Unitarian. I believe in God.
I do not believe in the Divinity of Christ, and there are many other of the postulates of the orthodox creed to which I cannot subscribe. I am not, however, a scoffer at religion but on the contrary recognize, in the fullest manner, the elevating influence that it has had and always will have in the history of mankind." (Emphasis added)
Taft's biographer Henry Pringle thinks that "it was well that political enemies did not have access to his private files" because the phrase "I do not believe in the Divinity of Christ" if made public "would have been more than enough to send Bryan to the White House in 1908."
The Life and Times of William Howard Taft, vol. 1 , p. 45.
https://archive.org/details/in.ernet.dli.2015.164430/page/n59
Would it, though? Suppose in the heat of the campaign the letter had somehow been made public--let's say some shocked janitor in Henry Taft's office comes across the letter and decides he must make it public to save America from the curse of an un-Christian president. Remember that Taft made no secret of the fact that he was a Unitarian, and of course many people denounced him for that and said that they would not vote for him precisely because as a Unitarian he did not believe in the divinity of Christ. But there were also no doubt people who had only a fuzzy idea of what Unitarianism was--who thought it was just another Protestant denomination--but would have been shocked at the phrase "I do not believe in the Divinity of Christ" even if in fact it was implicit in his Unitarianism. So I do not doubt that Taft would have lost
some votes (though there would also be some sympathy for him for the way in which a private letter of his was made public). But I doubt that it would have been enough to let Bryan win...