How much historical evidence outside DC is there for the existence of Batman? The only sources that seem to exist are from DC followers, but if he really existed then surely contemporary, non-DC historians would have mentioned him right? To me, the Batman Myth Theory seems spot on.
This point of view is laughably misinformed at best, and revisionist Arkham propaganda at worst. The fact is we have a multitude of sources indicating the existence of a figure known as Bruce Wayne he operates under the alias of "Batman" - in fact there is ample evidence to suggest that he may be operative today. Batman Myth Theories largely stem from the varied cult followings of Arkham Asylum - with followers flat out denying the existence, or at the very least misconstruing/underestimating the power of Batman.
The Batman Myth theory however is not completely without credibility - in particular the existence of one "Robin" is a hotly debated topic. Indeed in a magnificent series of documentaries about the subject from 2005-2012 (which include primary footage) the character presented leads us to believe the "DC" Robin is in fact a composite character, not entirely without purpose to Batmans life, but rather representing several people of minor importance to the Caped Crusader.
As a matter of fact, these documentaries are also our primary source regarding the elusive criminal organization "The League of Shadows". Interestingly here we see Nolans documentaries present Henri Ducard and Ra's Al Ghul as one man, as opposed to the conventional DC interpretation of two separate people - quite in contrast to the presentation of Robin.