In a world where tigers didn't exist, I wouldn't expect to see one in my trashcan. In a world where tigers did exist, I also wouldn't expect to see a tiger in my trashcan, but I wouldn't be quite as surprised if I did see one. My prior probability that tigers exist is very high, since I have lots of independent reasons to believe that they do exist. The conditional probability of observing no tiger in my trashcan is skewed very slightly towards the world where tigers do not exist, but not enough to affect a prior probability that is very close to 100% already. You could say the same for the goblin example, etc–my prior probability is close to zero, and although I'm more likely not to observe a goblin in my trashcan in the world where goblins don't exist, I'm also not likely to see one in the world where goblins do exist. The prior probability is far more skewed than the conditional probability, so the evidence of not observing a goblin doesn't affect my belief much.

The fact that you haven't seen a tiger in your trashcan is, however, evidence that there is no tiger in your trashcan.

Edit: Which I think is more or less harmonious with your original post. It appears to me, however, that at some step in the discussion, there was a leap of levels from "absence of evidence for goblins in the trashcan is evidence of absence of goblins from the trashcan" to "absence of evidence for goblins in the trashcan is evidence for the complete nonexistence of goblins".

For practical purposes, sure, this is a case where "absence of evidence is evidence of absence" is not a very useful refrain. The evidence is so weak that it's a waste of time to think about it. P(I see a tiger in my trashcan|Tigers exist) is very small, and not much higher than P(I see [hallucinate] a tiger in my trashcan|Tigers don't exist). A very small adjustment to P(Tigers exist), of which you already have very high confidence, isn't worth keeping track of... unless maybe you're systematically searching the world for tigers, by examining small regions one at a time, each no more likely to contain a tiger than your own trashcan. Then you really would want to keep track of that very small amount of evidence: if you round it down to no evidence at all, then even after searching the whole world, you'd still have no evidence about tigers!

It's not fully accurate to say

Only provided you have looked, and looked in the right place.

but it might be a useful heuristic. "Be mindful of the strength of evidence, not just its presence" would be more precise, because looking in the right place does provide a much higher likelihood ratio than not looking at all.

I don't think the fact that I haven't seen a tiger in my trashcan is evidence against the existence of tigers.

Is it because you deny that P(H | E) > P(H) in this case? Or do you acknowledge that P(H | ~E) < P(H) is true in this case, but you don't interpret it as meaning "the fact that I haven't seen a tiger in my trashcan is evidence against the existence of tigers."

If you deny that P(H | E) > P(H), this might be because your implicit prior knowledge already screens off E from H. Perhaps we should, following Jaynes, always keep track of your prior knowledge X. Then we should rewrite P(H | E) > P(H) as P(H | E & X) > P(H | X). But if your prior knowledge already includes, say, seeing tigers at the zoo, then the additional experience of seeing a tiger in your trashcan may not make tigers any more likely to exist. That is, you could have that P(H | E & X) = P(H | X).

In that case, if you've already seen tigers at the zoo, then their absence from your trashcan does not count as evidence against their existence.