Some LTW teachers require incomplete thoughts on the outline and some don't.
From the beginning (on the Rudimentary Persuasive Essay), we suggest you require students to do the following:
1. From ANI to OUTLINE: Translate any proofs in complete thought form into incomplete thought form
2. From OUTLINE to ESSAY: Translate proofs in incomplete thought form into complete thought form
Why? It isn't about style practice; it is about grammar assessment and teaching at the point of need.
If a student can easily translate proofs from complete thought form into incomplete thought form and then back again, he demonstrates that he understands the difference between a complete thought and an incomplete thought. If he cannot easily apply this foundational grammar knowledge, you can then begin to teach him what he needs to know about incomplete thoughts, complete thoughts, and sentences. That way when you write "fragment" on his paper, you know that he knows what you mean, and that he has the skills to correct his work.
The RPE is a perfect time to talk about the different ways one can turn a complete thought into an incomplete thought (a word, a phrase, a dependent clause, or an independent clause that does not make sense). The (not-so-simple) exercise of turning sentences into non-sentences is a perfect practice for assessing what students truly know about sentences.
After you're certain that a student can see a complete thought, can express in words what a complete thought is, and can easily apply "complete thoughtness" and "incomplete thoughtness", then you can allow him to write complete thoughts on the outline. "Incomplete thoughts on the outline" has served its purpose.
In other words the "incomplete thoughts on the outline thing" is about teaching grammar at the point of need.
Since this is the arrangement phase, our focus is on the argument itself, not on "saying it nicely." That comes next, in the elocution stage. There is also the possibility that once the student puts effort into complete sentence outlines, he will then transfer that into paragraph form and think "done!" Nope! You want them to concentrate on writing beautiful sentences and paragraphs, complete with the schemes and tropes we are learning.