Remember that MSE is meant to be a repository of questions and answers for everyone, not necessarily just the person who asks a question. There is room for a question to have answers at multiple levels, because askers at multiple levels may come across the question in the future. Hence the "goal" of an answer could be either to address the asker's current need, or to provide the most "elegant" possible answer. Both kinds of answers should exist, as both kinds of answers help future askers.
Regarding the voting on the cited question, I have noticed that "elegant" answers using more powerful tools tend to get more upvotes than more "elementary" answers (at least in analysis and in the related tags which I follow). I think that there are two large categories of upvotes, which account for a majority of the upvotes on these kinds of answers (and note that I am veering wildly into opinion here):
Aesthetics, in the sense that mathematicians often prefer to prove things at a higher level, in a more abstract manner, using fewer words. In a lot of ways, this makes sense—the whole reason to build up a body of theory is to allow for the use of powerful theorems which can make difficult results appear trivial. Hence some more advanced voters (PhD students, faculty in math departments, etc) may upvote more abstract arguments because they appeal to their sense of aesthetics.
A cargo cult mentality, in the sense that there are a lot of undergraduates who use this site who are impressed by the use of powerful tools to solve simple problems. For example, a user recently asked about a YouTube video which made the following claim
First, suppose that our initial chunk is part of a parabola, or if you like a cubic, or any polynomial. If I then tell you that my mystery function is a polynomial, there's always going to be exactly one polynomial that continues our initial chunk. In other words, a polynomial is completely determined by any part of it.
One answer was relatively elementary: if $f$ and $g$ are polynomials which agree on an infinite number of points (such as an interval), then $f-g$ is a polynomial, which is zero on an infinite number of points; but the number of zeros of a polynomial cannot exceed the degree of that polynomial, unless that polynomial is the zero polynomial. Therefore $f-g$ must be the zero polynomial, so $f = g$.
Another answer invoked the Identity Theorem from complex analysis. A commenter to that answer noted that this was like swatting a fly with a nuclear bomb (and I can't disagree). Yet, as I recall, this answer had more upvotes. I suspect that many of these upvotes came from folk who may have been introduced to the Identity Theorem via that answer. They were not familiar with it before, but upvoted out of a sense of awe or worship for the use of heavy tools.
NB: The first answer now has more upvotes.
On the other hand, answers which use more "elementary" techniques have disadvantages which often prevent them from being upvoted. They tend to be more involved and technical, as one is restricted to using less powerful tools. Because of this, they are often harder to read and/or tedious to follow.
For what it is worth, I tend to prefer more elementary answers to elementary questions.
A last comment: I don't think that we insist that students provide their attempts. Rather, we insist that askers (who may or may not be students) provide context for their questions. The goal of providing context is to ensure that terminology is being used consistently, to help set the level of an answer, and to help make a question more searchable for future askers.
One form of context is an attempt at a problem—in principle, such context helps determine where an asker is really stuck, and also weeds out a lot of cheating attempts. In practice, I think that this form of context tends to clutter questions and make them unclear. If I were king of the world, I would not regard "an attempt" as context, and would ask for more useful information (what book is the problem from? what theorems can you invoke? what level of answer are you expecting? etc).
