-"That's an old, useless grammar rule." It is an old grammar rule. It is not useless. -"Hanging prepositions are a very common thing in English, no longer considered bad grammar" That is true, placing the preposition at the end of the sentence is officially allowed and for good reason. Churchill's quote effectively illustrates how strict adherence to this rule can be awkward and confusing. Language should facilitate communication, not hinder it.
The rule was amended, not abolished. Now speakers of English can choose.
Churchill was being facetious. He would have written: “I would not have put up with that”. This has the triple virtue of grammatical correctness, natural sound, and conciseness. If you read Churchill’s writing you will see that splitting infinitives was the exception to the rule. Since DuoLingo permits multiple translations, I recommend accepting the grammatically correct translation as an acceptable answer. It is hard to see the downside.