Yes; it was part of a general project aimed at preventing popular movements from any sort of lasting power in American government. The idea that FDR was breaking any sort of tradition by running for a third term is entirely false and easy to debunk as well. The 22nd Amendment was an entirely novel idea.
No idea how much of this is true, but a point in its favor worth mentioning is that America, beyond the first thirty years or so, never really had regular two-term presidents until after the 22nd Amendment was passed. This isn't cause and effect--party rotation became more regularized, and leaders started living longer. The only three I can think of that served a full eight years and nothing but are Jackson, Grant, and Wilson, and at least one of these had third term aspirations. America was a nation of one-term, assassinated, or accidental presidents, combined with either quick party rotation in office or long periods of one-party presidential rule.
Grant and Wilson both did, with Grant actually seeking a third term at the 1880 RNC and Wilson hoping that the 1920 DNC would end up turning to him as a compromise option. (You also missed TR, who served virtually a full two terms, stepped aside in 1908 in accordance with tradition, but then regretted it for the rest of his life, seeking a third term as a third-party candidate in 1912 and being the initial frontrunner at the 1920 RNC before his untimely death).
Anyway, the tradition definitely existed before FDR but it was never absolute and many interpreted it as a two-consecutive-term limit. The idea that the 22nd Amendment happened as part of a backlash to FDR's policies is true and important, I think; there was a point in the late 1940s/early 1950s when really anti-New Deal politics were quite close to taking power, and it's really the personality of Dwight D. Eisenhower which stopped them.
