Question

Global class field theory. Statements are relatively simple (of the form "the abelianisation of the absolute Galois group of a number field (or, more generally, a global field) looks like this"). This is only perhaps 90 or so years old---it all depends on exactly what you mean by global class field theory. I'm no historian, but precursors to the theorems in their current form are over 100 years old and Hilbert already raised the question of making them more "explicit" as one of his problems (he wanted to see concrete generation of abelian extensions of an arbitrary number field rather than an abstract isomorphism of of a Galois group with another group). Original proofs were surely longer than current proofs, but current proofs are still very very long. If you want a proof of the main theorems of class field theory then there are nowadays a number of books which you can choose from. I think this is a strong contender for hard evidence that the belief in the original question is way too optimistic. I think that things will, in some sense, only get worse!