At the risk of getting political, I absolutely cannot imagine, say, Barack Obama, George W. Bush, or Bill Clinton getting minted on highly-circulated coinage (and not just commemorative editions) without an absolute tidal wave of controversy. Doing my own quick reading appears to reveal controversies (in the case of the dime) relating to communist conspiracy in the initialing on the dime, as well as controversy over the potential plagiarism of the FDR bust featured, but nothing specifically about the depiction of FDR himself. I'm especially intrigued by this because FDR had just come off an unprecedented four-term presidency which must have come with controversies of its own.
Were these presidents just so overwhelmingly popular that they warranted so fast a mint? Or does this suggest something not about the Presidents themselves, but about the American political climate of the 20th-21st century? Is it a special case because they had died prior to the minting, no matter how soon after the mint came?
What controversy existed about these depictions just a year after these Presidents were gone, and can any meaningful historical insights be gleaned from that controversy, or lack thereof?
Both FDR and JFK were presidents who died in office, in a time when the level of respect in national discourse was higher. Unfortunately, I cannot speak to the FDR dime question, though I would note the "Stalinist initials controversy" you mentioned with the dime also affected the Franklin half dollar, which also features in the story of the Kennedy half-dollar.
Due to the national outpouring of grief at Kennedy's assassination, there was not only no partisan opposition, it was in fact essentially unanimous, and the Kennedy half-dollar's "origin story" is rather interesting, with the whole process taking a matter of weeks, which is not normal.
But first, a bit of a primer of American coinage. The important thing to remember is that the trend of using American statemen on coins was very much a product of the 20th century, with the Lincoln penny introduced in 1909 and Washington and Jefferson following in the 1930s. Prior to that era, American coinage consisted of more metaphorical designs such as personifications of Liberty, Peace, the god Mercury, wheat pennies and the "Indian head"/buffalo nickel.
Eva Adams, the director of the US mint, began directing her subordinates to explore a Kennedy coin within hours of Kennedy's death. The design was able to be completed quickly by working off prior art, namely the commemorative US Mint medal produced for his inauguration.
From a numismatic point of view, the Kennedy half-dollar is noteworthy because it cut short the production of the Franklin half (making Franklins much rarer), which is partly how we know more about how the Kennedy half came about. The reason for this is being that Federal law at the time prevented a coin design from being replaced prior to 25 years without an act of Congress. The Franklin half had been introduced in 1948, and at the time of Kennedy's death in 1963 had only been in production for 16 years. There was some discussion at the time of putting Kennedy on the quarter, but Jackie Kennedy preferred the half dollar.
President Johnson announced the new coin on December 10th, the design was finalized and approved by the Treasury Secretary (with Jackie Kennedy's approval) on December 22. A joint Congressional resolution (H.R. 9413) authorizing the replacement of the Franklin half dollar coin was passed on Dec. 30. The bill was introduced by Henry B. Gonzalez of Texas, who was in the presidential motorcade in Dallas, and had helped carry JFK's casket to the hearse that took it to Air Force One, and went on to serve as chair of the House Select Committee on Assassinations.
Dies were delivered to the Mint to begin coining three days later, just over a month after Kennedy's death.
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