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.