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.