Why the "burden of proof is on the theist" argument should go the way of the dodo

Only a hundred years after the dodo was officially declared extinct, Europeans refused to believe such a fantastical creature had existed.  It became a myth much like the unicorn.  By 1800, professional naturalists were casting doubt on written descriptions of the bird, as well as on extant drawings, many of which did display striking inconsistencies.  It became scientific vogue to deny the bird's existence and to challenge the Oxford head and foot [the "remains of the last known stuffed dodo"] as fakes.  Why, skeptics asked, has no one preserved a skeleton?  Skepticism was so strong by 1850 that a group of zoologists journeyed to Mauritius in search of dodo bones.  [After none] were found..., the dodo was denounced as a scientific fraud.
Finally, in 1863..., a resident of Mauritius... discovered numerous scattered bones that he painstakingly assembled into several complete dodo skeletons [and] shipped [them] to major museums where they were studied and pronounced authentic. Thanks to his efforts, we not only know with certainty that the dodo existed, but what it looked like.

-Charles Panati, p. 203-204

Moral of the story: simply because something can't be proven doesn't mean it doesn't exist, a.k.a., lack of evidence isn't necessarily evidence of lack, a.k.a. "There are [at least potentially] more things in heaven and earth Horatio than dreamt of in your philosophy".  The unspoken assumption in the "God's existence must be proven in order to be worthy of belief" argument is that everything in existence must be provable.  Yet, it would have been those who argued that the dodo existed who would have been right despite their inability to prove this.  Likewise, the lack of proof for God's existence is not sufficient to carry the argument.
Complicating the situation is the supernatural nature of God, assuming He (1) exists.  Since our senses can normally only detect material objects, we lack the means by which
Returning once again to the debate over the dodo, each could muster reasons to support their position.  The pro-dodo camp could point to the preserved head and foot, the skeptics could counter with the lack of a complete skeleton.  If the former tried to buttress their argument by referring to the drawings made of the dodo, the latter could try to discredit them as evidence due to the inconsistencies.  The problem is that there is no known way to objectively weigh the strengths of each side's arguments.  Which should count more, the head and foot and drawings on the one side or the lack of a skeleton and inconsistency of the drawings on the other?  A similar predicament arises in the debate over God.  A given proof considered as airtight by one side is treated as inconsequential by the other.
Further complicating the matter is that our senses can only normally detect material objects.  God, however (assuming that He exists) would be a supernatural Being.  Therefore, we could only detect Him if He permits us to.  We lack the means to force Him to do so.  Thus, a God who we must be able to prove would be a contradiction in terms.
All this probably could have been avoided in the beginning if theists had, instead of taking up the atheist's challenge to prove the existence of God, questioned why they should have to.  After all, almost everyone accepts at least a few beliefs as true without proof and it would be nigh impossible to get through daily life if they didn't.  Imagine, for example, if one had to prove that his or her food was not poisonous before eating it.  Oddly enough, support for the idea that proof is too high a standard comes from Austin Cline, About.com's Agnosticism / Atheism Guide, who writes: "The first thing to keep in mind is that the phrase 'burden of proof' is a bit more extreme than what is often needed in reality. Using that phrase makes it sound like a person has to definitely prove, beyond a doubt, that something is true; that, however, is only rarely the case. A more accurate label would be a 'burden of support' - the key is that a person must support what they are saying."  A better measure, I contend, would be that as long as I believe the arguments for God's existence and against His non-existence outweigh the arguments against His existence and for His non-existence, it is rational for me to continue belief in His existence.  On the contrary, it would be illogical for me not  to believe in His existence in such a case.
(1) I shall, for the sake of convenience, refer to such a Being using the traditional Judeo-Christian masculine forms

N.B.: I realize that, unlike the dodo, God, if such a being exists and chose to do so, could reveal Himself, but that's a topic that will be addressed in a whole 'nother (future) post.

Panati, Charles.  Panati's Extraordinary Endings of Practically Everything and Everything.  New York: Harper & Row (1989).