Should one stick to the organization of a paper while teaching a subject? When teaching a specific topic in a subject, which is not generally covered by textbooks, should or when the instructor should stick with the structure of the paper?

A not very popular topic in computer science covered in a paper. The structure of the paper is Algorithm 1 - Lemma 1 - Algorithm 2 - Lemma 2 - Theorem 1 - Algorithm 3.

The paper is published in one of top journals. I am aware that the editors and reviewers are quite decent people and they really know how to suggest organization of the paper for the sake of readability and understandability.

In what cases should an instructor change the structure while teaching, for instance to Lemma 1 - Lemma 2 - Theorem 1 - Algorithm 1,2,3? 

Answers with experiences will be appreciated.
Even if the authors are excellent writers, there's no reason to assume that the organization the authors chose as best for an audience of specialists is also the best one for your class.  (At least in math, it's very rare for the editors or reviewers to call for organizational changes.)

I'd feel quite free to present the material differently if I thought a different order emphasized the points most relevant to my students, would be easier for them to follow, or I just preferred it. Indeed, I think there's some benefit to intentionally giving a different structure, since I think students benefit more from having access to multiple explanations than from repeating one slightly better one.