I have seen other schools and extracurricular programs aspire to include math "research" at younger levels without really understanding what that entails. Many programs don't understand that this usually doesn't happen until the end of a doctorate.
Also (as you brought up in the comments below), focusing on the part where one just sets up the research question is not a good way to simplify the task because that part is typically done by the senior researcher or a doctoral student's advisor.
I think that a modified version of this might be a valuable activity for very advanced and motivated students. As written, I agree that it would be excessive to require every student to do the program you described. I also think it's unrealistic to expect every teacher of being qualified to carry it out. An alternative I would suggest is to first have the teacher find a handful of math papers that seem accessible (I would look only at expository journals, not research journals), then have a student pick one of those papers, and the project would simply be to read it (very carefully, over a long time), and write a review of it explaining the idea and what they think about it.
You also asked about explaining the distinction between math and other disciplines when it comes to research. I agree that is probably the source of the problem here. I'm including some comments on that referencing the items from your list.
- "Clarify by conducting a literature or artifact review or developing a main argument."
Math research literature is incredibly specialized. Not only does one usually need an advanced degree to be able to read it, but one needs to have specialized in a similar area as the author. And if we are talking about literature pertaining to open problems, it would be very difficult for a student (probably even the teacher) to do this review.
Evaluate the information by identifying:
-the credibility of sources
-feedback from peers and professionals in the field to improve the proposal/project
In math, the credibility of an article is provided by whether or not it has been published in a peer reviewed journal. The math research community has a rigorous revision and refereeing process for articles before publication. So I would interpret this part as: restricting the student's sources to peer reviewed journals, as opposed to Wikipedia, blogs, or other information. But I think it would be more appropriate for students to be looking at those "less credible" sources because they are friendlier and potentially accessible.
- Organize the presented content and/or project timeline in a manageable, usable, and discipline appropriate format that can result
in implementation of the research proposal or project.
Nobody, not even seasoned research mathematicians, can foresee the twists and turns that happen along the process of making a new mathematical discovery. For example, sometimes some trivial thing you overlooked at first turns out to be significant enough that it gets its own paper. It's not uncommon for a doctoral student to think he's a year away from graduating, then run into a theoretical problem that pushes that back a couple of years (or even indefinitely). The path is very unpredictable by nature.
Critically analyze the strengths and weakness of the information collected, including, but not limited to:
-supporting and opposing arguments
-the proposed process
There are no supporting or opposing arguments among proven math theorems. It's not like the social sciences where we are relying on evidence. We are relying on proofs of abstract logical statements. (There are some cases where mistakes end up published, but that's not what we're talking about here.) A known, checked math result has the consensus of the mathematical community, making this step basically irrelevant.
- Synthesize by constructing or summarizing all information/data into a research proposal/project that includes rigorous, researchable
questions based on new understandings.
That is essentially what one does as a doctoral student after completing all their qualifying exams, and even then it is done under the guidance of a professor.
In the end, I would be happy that the school is expressing interest in something like this, but I would try to (in a friendly way) gear them toward something more realistic.