America has a long tradition of strong anti-monarchism with the tradition of 2 terms being strongly enforced ever since Washington. Did the republican opposition during the elections of 1940 and 1944 make this an important talking point? And why was FDR still able to continue his presidency, while after him the 2-term rule stayed in american political tradition?
More can always be written, so while you're waiting on an answer, you may be interested in three previous posts of mine:
This post details the circumstances of Washington's decision not to run for a third term. Summary: it's arguable that he intended to set any kind of precedent. It was Jefferson who was explicit about presidents not serving more than two terms.
This post details how strictly the precedent was actually upheld. Summary: Not very strictly. Several two-term presidents before FDR had explored the possibility of a third term. Few two-term presidents who survived to the end of their second term hadn't explored the possibility.
This post details the circumstances surrounding the passage of the 22nd Amendment that enacted presidential term limits. Summary: it was certainly passed, in part, for partisan reasons, though it was not a particularly contentious issue. The belief that presidential term limits were a good idea went all the way back to Jackson and Jefferson. Once FDR made the possibility a reality, the Republicans moved to enact term limits, with the aid of like-minded Democrats to get the amendment passed. Your question is somewhat directly addressed here: yes, there was criticism by the Republican Party during the 1940 and 44 elections against third (and then fourth) terms, because they could lead to anti-democratic government.