I don't doubt his legitimacy as President. However, I do grow suspicious of someone who refuses to give up items that require little to no effort.
As far as the magnifying glass being on you leading up to and during the presidency, it's an occupational hazard. It's part of accepting the position of a high profile politician. Everything you do, and everything you've ever done, will be under intense scrutiny. That's just always the way it will be. Of course, there will always be bogus claims, but if disproving those making a ruckus would be so easy, why wouldn't you???