How important/necessary is it to find or learn an elementary proof? There are a lot of theorems and results in mathematics that are very easy to state but often require a lot of advanced machinery to prove. We could easily come up with many examples of this: Dirichlet theorem, prime number theorem, and many other results in number theory, existence and regularity results of some PDEs, and the classification of, for example, finite groups using difficult representation theory. However, some problems that are solved using modern methods turns out to have an elementary solutions as well.
When we try to see how a theory is developed from axioms and assumptions, we often need to be more careful when using some "advanced" results, since we can only use things we have already proven; in other words, we need to avoid circular arguments. But now, let's consider a different scenario: problem solving. When we try to solve a problem, like things similar to Dirichlet theorem, we are applying theories rather than reconstructing them from axioms. In this case, how useful is it to solve a problem using only limited amount of machinery or even with elementary methods?
The question "how useful" might be difficult to judge,  so actually, any ideas regarding why we need/needn't learn elementary proofs are helpful.
 A: A method of proof that you have not learned is a method that you can not use yourself. It would be best to learn all the different methods of proof. It is provable in mathematics that not all true theorems will have a proof. It follows then based on that axiom that not all true theorems have an elementary proof. This is a good motivation to never try to establish proofs for difficult problems. But that way of thinking leads to a pseudo science that has no proofs. Therefor we must balance or gamble our time looking for proofs and not looking for proofs. Maybe we flip a coin or take turns role playing good science bad science. I think it is important to remember that the definition of an axiom is something that is intentionally given as true without proof. This is the basis for all proofs that rely on axioms. Some axioms are provable with other axioms. This is another way to say that mathematics as it is defined by general consensus is consistent.
In my opinion the majority of the good work done in mathematics is accepted with proof and refuted without proof. 200 years ago it was common for new discoveries published with proof to be rejected by peers that did not understand or believe the proof or the subject matter of the research. This is how galois group theory was at first rejected and buried. A case study in how mid-wits working in academic institutions rejected intelligent forward thinking mathematics. This is one of the problems that lingers on today to a much lesser extent. Elementary proofs are really good for getting a paper passed through peer review where it would otherwise maybe be rejected even if the math was valid. Terry Tao's recent collatz paper comes from UCLA but it was targeted towards a 6th grade level elementary student. it is also significant that there is absolutely no proof of the collatz conjecture anywhere in the paper. It is interesting but it is at most a secondary conjecture to the original Collatz conjecture. Remember that this is exactly the thinking in why every other paper about the Collatz conjecture was rejected. But with the cult of Terry Tao and UCLA, anything gets through. I still think He is one of the greatest mathematicians alive today but that should not be an excuse for a double standard of proof. In my opinion mathematics should be consistent if it is not pseudo science.
Perhaps the most important property of a good theory is that it is consistent over time more so then it is consistent with pre-existing axioms. According to Steven Wolfram and others, there are an infinite number of systems of mathematics that have axioms different from our current axioms but that they are still valid and consistent within themselves. This is stated without proof but it is also a meta observation about proof.
