If Churchill was trying to show that the rule is silly, then he stacked the deck when he wrote "This is the sort of [stuff] up with which I will not put." That sentence simply doesn't prove his case. It's awkward only because he deliberately made it awkward. He could have written, "I will not put up with this sort of [stuff]."

If you ever have to choose between clarity and good grammar, choose clarity – but you seldom have to choose. You can nearly always have both, if you think about it a little bit.

And Leeniea is right that most of the "prepositions" in that other sentence ("What did you bring that book I didn't want to be read to out of up for?") are actually adverbs. A preposition always has an object. If I say "Bring the book up the stairs" then "up" is a preposition and "stairs" is its object. But if I say merely, "Bring the book up," then "up" is an adverb.

I don't do this sort of thing for a living, but I wish I could.