I think the more realistic POD is not the Mob going along with Flegenheimer on this but his killing Dewey before they can kill Flegenheimer. I discussed this many years ago in soc.history.what-if:
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Supposedly one of the reasons Arthur Flegenheimer (better known as "Dutch Schultz") was killed was that he was attempting to have the young prosecutor Thomas E. Dewey murdered--something that the rest of the Mob thought was a very bad idea, likely only to result in public outcry and a severe crackdown on crime.
Anyway, suppose Schultz had succeeded (before himself getting hit, either for the Dewey job or for other reasons) in killing Dewey. In that case, (a) Who becomes the GOP candidate for governor of New York in 1938 (I assume whoever it is will be defeated by Governor Herbert Lehman as Dewey himself was that year) and 1942 (when Dewey succeeded)? (b) More important, who are the likely Republican presidential candidates for 1944 and 1948?
Of course, (a) and (b) are not unrelated. In addition to all the OTL rivals of Dewey for the nomination in 1944 and 1948, any Republican who succeeded in getting himself elected governor of New York in 1942 and 1946--and there is no reason to think that only Dewey would be capable of this, both of these having been good years for the GOP--would instantly be mentioned as a candidate for the GOP presidential nomination. Maybe Irving Ives (1896-1962) who in OTL was a state legislator from 1930 until he was elected to the US Senate in 1946. (Ives was re-elected in 1952; in 1954, upon Dewey's deciding not to run for a fourth term as governor, Ives became the GOP candidate for that office and lost narrowly to Averell Harriman...)
(Naturally, another consequence of Dewey's murder would be a tremendous demand for a crackdown on organized crime--though how long such a crackdown would last is another matter...)