Question for this post: In my post, how do I make the motivation and question rigorous and readable to get a proper answer. (It received downvotes when I first originally sent it to math overflow but I believe if it’s asked properly it can be answered here or there).
At the time of writing, your question has one answer, also from yourself, where you've begun with:
I'm not finished.
I am posting my attempt here instead of the post since the answer is too long.
I think this must be an indication that the question as written is too broad.
There is always a chance that a query will match up with some known thing in the literature, and someone will say "ah, yes, this is Dave's Theorem" and give you a brief, satisfying answer. But it will also often be the case that a great deal of work or insight is needed in order to connect known results to the query at hand. Random Internet volunteers are not necessarily inclined to do that work so there is a probable bias against engaging with a question that "smells like" it will be too much effort to try to answer.
Some of those smells on a question in general might be if the question is very long, or has lots of sub-questions. In your case, even before trying to read or understand the mathematics, we see that there is a "main question" which itself entails several different cases ("linear or superlinear", "...or a method", what is a "natural" choice function), followed by explicit subquestions of which at least part (4) seems to be strictly beyond the scope of the main question. It also refers transitively to a couple of previous questions, which enhances the impression that there is a lot of context to consider. Overall, this feels like a more open research question, where you have a hunch that something like the thing you want might possibly exist, and you are looking for assistance in navigating the conceptual space around it. You have a sense that there ought to be a "meaningful" extension, and some idea about how to make "meaningful" more precise, and you're asking for a great deal: for someone to come up with the proper criterion, and a construction meeting the criterion, and some further analysis.
This site is a better fit for questions that are more tightly scoped, so you should begin by trying to pull out some precise and answerable question out of the progress you have made so far. It is perfectly possible that this site is not going to be much help for you, compared to any of the other resources available in the universe. But if you do have a more specific version of a question, you could then focus on presenting it clearly for the purposes of this site, and that's when we would get into detailed issues of readability and rigour.