Even if this anecdote isn't true (and several variations on Churchill's purported reply are floating around), the comment illustrates how the rule can be silly. English speakers end sentences with prepositions all the time, and it's often odd to do otherwise. You might ask someone, "Who did you give it to?" (or if you're a stickler, "Whom did you give it to?"), but you would never, ever say, "To whom did you give it?" unless you wanted someone to make fun of you.
Still, The Awkward Adverb believes that following the rule, when possible, lends elegance and clarity to formal writing. And Churchill wasn't being entirely fair. The motivation behind the grammatical principle is to keep prepositional phrases intact, and Churchill's sentence doesn't even include a prepositional phrase. "Put up with" is a verbal unit that means the same as "tolerate."
So he could have scrawled, "This is the sort of English that I will not tolerate." This response isn't as intentionally awkward, but it's also less funny.