Yes, actually, the burden of proof in the scientific process is on the person making the claim. The scientific process is, by its very nature, a skeptical one - it isn't "I want to believe" it is "I am willing to believe." They may seem, initially, to be similar statements, but there are undertones to each that are dramatically different. When you say "I want to believe" you are starting out from an initial position of belief in a claim regardless of supporting or refuting evidence; conversely, "I am willing to believe" should imply that certain conditions must be met in order for the claim to have merit. So to make a scientific claim one must also provide evidence that supports the claim, show how the evidence supports the claim AND show how the evidence DOESN'T support an alternative claim. Science doesn't conclusively prove things to be true, it proves things to be conclusively false. When you start from a position of "I want to believe" you start at the conclusion and work your evidence to support your claim. In fact, your open mindedness is inversely proportional to your want for believing. If you start from a conclusion and work your way backwards to a claim you must be willing to reject any evidence to the contrary of your conclusion. You have already implicitly stated that you are unwilling to accept explanations that refute your claim.
So, yes, the burden of proof is, in fact, on you as the claimant, but not only must you provide evidence to support your claim and demonstrate how that evidence supports your claim, you must also demonstrate how that evidence isn't the cause of some other phenomena.
And if you would like to maintain your courtroom analogy, then we can do this scientifically, anecdotal evidence carries little to no weight in the scientific process. Your stories will not pass muster, nor will unreliable source material. And I am sorry, but without those you do not do not have much of a case. Maybe you should have carried this out in a courtroom.