BTW, Coolidge's candidacy would be opposed by some as a violation of the no-third-term tradition. As I wrote here a few years ago:
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I discovered a very interesting thing in reading *The Third-Term Tradition: Its Rise and Collapse in American Politics* [the "collapse" part was obviously premature--the book was published in 1943] by Charles W. Stein: During the mid- to late 1920's, when it was discussed whether President Coolidge would run again in 1928--and many people thought he would, even after the famous "I do not choose to run" statement--the question was usually put as whether Coolidge would seek a "third term" in 1928. There was much debate as to whether the anti-third-term tradition would stand in his way.
Now at first that surprised me, because Coolidge had served only nineteen months of Harding's term. But then it occurred to me: I was approaching the matter wrongly because I was unconsciously assuming that the Twenty-Second Amendment's definition of the anti-third-term rule ("for more than two years of a term for which some other person was elected President" etc.) had already been agreed to. But of course it hadn't been! The Amendment was in the distant future; the anti-third-term tradition was just that--a *tradition*, with no law to define its extent. Hence, people could and did differ on what a "third term" was. (At least prior to Grant's failure to get the GOP nomination in 1880, one could argue that the tradition only prohibited three *consecutive* terms--and of course TR tried to revive that interpretation in 1912, one problem being that it seemed to clash with what he himself had said on Election Night in 1904. [1]) Some argued that the whole no-third-term tradition was silly and should be scrapped, anyway. Others said that the tradition was sacred and should prevent Coolidge from seeking a third term. What would today seem an obvious position--that the tradition was valid, but that Coolidge had not served enough of Harding's term for it to be applicable--was indeed taken by some commentators, but was not quite as widely accepted as might be imagined. Incidentally, one advocate of another term for Coolidge proposed that he should promise if elected to resign on August 2, 1931, so that he would not have served more than eight years!
This controversy raises a question: If Coolidge ran and won in 1928--and let's assume for present purposes that his administration is not too different from Hoover's and that FDR gets in as in OTL in 1932--how would that affect the third term issue in 1940? FDR's supporters would say that the no-third-term tradition had *already* been breached by Coolidge and was therefore no longer valid. OTOH, it is possible that FDR himself would say something in 1928 criticizing Coolidge for seeking a third term which the GOP would quote against him in 1940. (In the same way, opponents of a Coolidge candidacy in 1928 searched old newspapers to see if Coolidge had said anything against TR's attempt at a third term in 1912. Alas, they found nothing--evidently Cal had remained silent...) But probably not to much effect. Almost everyone who voted for FDR in OTL would still do so--and would say that if FDR seemed inconsistent, well, after all, Adolf Hitler wasn't much of a threat to the world in 1928.
No doubt in this ATL it will later be argued that the language of the Twenty Second Amendment was adopted by conservatives precisely in order to attack FDR's legacy without casting aspersions on Coolidge's...
[1] "On the 4th of March next I shall have served three and one-half years, and this three and one-half years constitutes my first term. The wise custom which limits the President to two terms regards the substance and not the form. Under no circumstances will I be a candidate for or accept another nomination." Note that he didn't say "two consecutive terms" or qualify his "another nomination" with "in 1908." (Of course his supporters argued that, as the *Outlook* put it, "When a man says at breakfast in the morning, 'no thank you, I will not take any more coffee,' it does not mean that he will not take any more coffee to-morrow morning, or next week, or next month, or next year.")