There is a standard $3x3$ lemma in triangulated categories. See for example Lemma 2.6 in \url{math.uchicago.edu/~may/PAPERS/AddJan01.pdf}.

What a terrible talk, beginning with a terrible typo. That talk was when I was trying to work out an alternative construction of Voevodsky's Steenrod operations in motivic cohomology. The attempt failed (see the last few pages). I'd still like to see an operadic construction of those operations, but this is not the place to explain the difficulties I was seeing then. There are surely examples that others will know off the top of their head, but they are not on the top of mine.

I agree. I gave a new proof of the splitting priniciple in math.uchicago.edu/~may/PAPERS/Split.pdf that emphasizes that point of view.

@LSpice Thanks, Loren. I did need a laugh.