Oh, I can well understand this position and inclination. But you are confusing this normal human reaction with Science.
A scientific investigation of the super-natural, the miraculous, the non-lawful natural, is not known to come up with anything reliable. How many good science discoveries have you heard about regarding ESP and the Paranormal?
"To use a previous example, if science only investigates the lawful conduct of natural law then it could not have studied the unlawful (according to Newton's laws) precession of Mercury's orbit."
You seem to be confusing the "observation of non-lawful" behavior with the Science of non-lawful behavior.
Science is about uncovering the lawfulness of something ... and until then, all Science has is a chaos of observations and "facts".
Certainly Science noted that there was a Mercury.
Certainly Science noted that Mercury's movement did not follow a known rule very well.
But this is not the completion of the Scientific endeavor.
Mercury's orbit didn't become "Good Science" until someone developed the correct hypothesis for what was affecting Mercury's orbit.
So.. if a man came to your town (wearing a badger suit) - - and he said he was God - - would you believe him?
If someone leaned towards you and whispered, I think he's from Mars ... how would you use science to determine whether he was a Martian or God?
I think the odds are ... he would be from Mars.... how would you ever prove him to be God?