Now I ask you, does that really sound more natural than “That’s the kind of errant pedantry I will not put up with?”
No, obviously not.
Scholars are mainly the ones who fall for this baloney, but sometimes people who were traumatized by a misguided English composition teacher early in life will cling to it also.
Truth: There is no grammatical rule forbidding the ending of sentences with prepositions. But rhetoricians dislike such constructions because they make the end of the sentence feel weak. If I were inclined to edit the version attributed to Sir Winston, I would put “I will not put up with that kind of errant pedantry.” See? Ending with “errant pedantry” is strong.
But I’m not inclined to edit it, because it is funny.
*—There's no evidence to back up that story, but it does provide a useful example.