Should proofs include a third “context” column?

I disagree. Two column proofs of the sort introduced in American middle school and high school mathematics programs are really just an attempt to introduce students to the beginnings of formal mathematical proofs and show that there is a world beyond prose proofs. No mathematicians actually use two column proofs (and even formal proof systems don't use two column proof notation). The state of formal mathematical proof technology is not currently at the stage where it makes sense for even professional mathematicians to be writing formal proofs, hence it's not even clear that two column proofs have a place in a future world where (hopefully) formal proofs become the norm in theoretical mathematics, depending on how formal proofs ultimately become structured. And the "third column" is effectively just comments, which are very easy to bolt on to any system after the fact.

As it is right now, it is enough to know merely how in theory a prose proof could be reduced to a formal proof. And besides most middle school and high school students aren't going to be able to write even a rigorous prose proof. Trying to stack more stuff on the edifice of two column proofs seems like just saddling students with even more (from their point of view) tedious make-work.

Is this ``two-column'' proof style an American thing?

I'm not a mathematician but I've written my fair share of proofs (incl. math competitions) and I don't ever recall being taught or having used something like this.

No, they should be written in comprehensible and elegant prose that is as short as possible while allowing an informed reader to check each step.

I don't know at what level this is used, but I have only ever seen it in my graduate level logic and mathematics courses. (I'm European.)

I like it. Part of a good proof is a rigorous derivation. There is plenty of thing sto be said about what intuition can lead us towards such a proof. But at the end of the day, a good proof is ideally almost machine-checkable. Most proofs (much like code) lives two lives: the one in which they are written, and the one in which they are read. It should be almost trivial for a student to go back, on their own, days or weeks later, and see that each step is valid. And I think prose interspersed between column-style proofs do a better job than what margin notes do. The reason two columns work well is that the step and it's justification are of both condensed and almost always of similar length. Justification (much like code comments) may or may not need a good chunk of space. And surely nothing is stopping anyone from inserting little margin notes here and there in a two-colum proof, right?

I love reducing a problem to its minimal formalism.

I hate documentation that only includes the minimal formalism without examples.

It suggests the author never developed “Theory of Mind”.