Are simplified elementary proofs if valid interesting to the professional mathematical community For the last 10+ years, as a math amateur, I worked nightly on understanding the distribution of primes and the classic results in the history of Fermat's Last Theorem. 
I have made numerous mistakes and for the last seven of those 10+ years, I have, at a rate of once per year, submitted what appeared to me to be valid proofs to the math.stackexchange.com community.  Up until this year, without exception, they were not valid.  In each case, a mistake was found indicating to me that I had much to learn and reminding me to be very thorough before believing that one of my arguments was worthwhile to propose to math.stackexchange.com.
This past week, I have a proposed an argument that was accepted on this web site and a more ambitious argument that I am waiting for evaluation where I expect that a mistake has been made.
It occurred to me that the work I am doing may not have any interest to the professional math community.  If someone were to solve a problem such as Legendre's Conjecture using surprisingly simple methods such as arguments from modular arithmetic and standard high school algebraic equations, would this be at all interesting? 
Would it be interesting if it greatly simplified a problem that had previously been given a state-of-the-art analytic number theory solution?
In other words, is the professional mathematical community interested in proofs that are valid but which do not extend the methods of mathematical analysis?
 A: Since you ask, I'm going to offer my opinion, but it is for you to decide how much to consider it. I am going to approach your question as: "I enjoy researching these topics , at the same time I am discouraged by the outcome. Would anyone even care if a simplified proof was found? "
My answer would be: Yes ... But.
Yes, A new proof may give new insights and perhaps even open new areas. Even if it does not, it is a joy to find and to see. And just delving into the topic may be rewarding for you if you enjoy it. So at the most positive, it could be enjoyable to try and a new proof is a permanent contribution to mathematics.
But How would you feel about the thrill of the chase without ever making a new contribution? You might be fine with it. if not, evaluate the odds. And, if you have been producing work which was not valid without realizing it, that is something to work on if you want to keep it up.
Do you know examples of 
Someone solving a problem such as Legendre's Conjecture using surprisingly simple methods such as arguments from modular arithmetic and standard high school algebraic equations
And if so, what was the case about the people who found them? 
Since you say that you have several attempted proofs with flaws in them and anticipate a flaw in your latest, I'll close with this advice: If this is something you like to do then it sounds like you could strengthen your proof ability. The usual way to do that it to take a course (or several) or work through books with good problems. And that is an activity which gives the joy of doing math. 
A: I think such work is important and has its place.  However, it is not clear that stackexchange is that place.
There are a lot of ideas to be considered.  Where simple algebraic and combinatorial methods are concerned, many of those have been tried and published or discarded.  However, such arguments have an educational value in teaching a student the limitations of such a method and giving that student an appreciation for the subtleties that arise in the subject.  I think it is good to have a repository of such ideas to use in future proofs.
Should such proofs appear on Stackexchange? It depends on the community taste.  I think very very short proofs (such as the sketch I gave of your argument re Sylvester Schur ) fit within the guidelines for MathOverflow, especially if you are asking about one step in the sketch, but the mandate for the forum is for discussion of professional mathematics, and there needs to be more to be a good fit.  Your question was good because it asked about extensions, and my answer provided a reference which is of interest to the research community: anyone else asking about an extension to S-S will see your post, my answer, and the valuable link provided.  
If you find a proof, post it somewhere on the web so that those interested will find a link.  Unless it is exceptional and enlightening, its audience will not be professional mathematicians, but others who wish to study and do mathematics.
Gerhard "Speaking As Doer Of Mathematics" Paseman, 2017.02.21.
