As a matter of logic, science being able to investigate the supernatural does not mean that science is able to investigate *any* supernatural claim, or some supernatural claim in particular, like "God exists." Do you agree? If not, where is the problem in the logic?
The problems are two fold.
1/ Supernatural is a subjective definition. Natural "cause" is a misnomer.
If it happens in nature it is natural. Including God , or any other being, acting in nature, if or when He acts in nature!
Natural "cause" is not cause at all. It is simply an observation that the universe is doing what it usually does. For example - the law of gravity is observation. It behaves as inverse square law. It does not say why gravity exists, or why gravity acts with the pattern it does. We do not know what gravity "is" or what CAUSES the effect we call gravity. We can only model how it normally behaves. ( space time warp is simply another canonical way of describing the math of observation)
So it is not a cause , it is an observation, of the universe doing what it usually does . It is also about what projects into our limited senses! Stuff can exist that does not (normally) impact our senses. The fact an expermental law is valid for the data set so far, does not mean either it has always done it, or will always do it either! Indeed - round galaxy shapes the math does not work! we call the error dark matter, which is a high sounding name for the math doesnt work!
2/ Science is axiomatic limited to what man put into it. The model can only declare apple or orange, if they are already defined . Since "godness" has no definition, the model cannot conclude God, whether or not God is real or caused a behaviour.
So all you can actually say in testing extraordinary behaviour is...
a- it is not doing what it normally does
b- it is violating fundamental axioms of the existing model, and cannot be reconciled without scrapping parts of the model and starting again.
( this requirement is to challenge those who say ... it is only a matter of time before science "iexplains" it eg Take the "time arrow" on prophecy, so far ahead chaos prevents prediction, and so unusual it cannot be guessed.)
-c that it does so in a context associated with Christianity.
That is all you can determine. A limitation of science not God. The rest is belief.
Incidentally - There is PLENTY that meets those tests..
Do they "prove" God? no - science cannot do that, without "godness" defined in the model science cannot declare it.
There may yet be a test for "jesus christness" (the man not God) , - watch this space!
They are "evidence" not proof.
Can you say all "ordinary" behaviour is therefore not driven by God? No... cant do that either!