if you make an event improbable enough, why shouldn't I be skeptical even if I think I observed it?
You should. You should be aware that you might e.g. have made a mistake and slightly misremembered (or miscopied, etc.) the results of the coin flips, for instance.
we would, in fact, find some sequences unbelievable
We might say that. We might even think it. But what we ought to mean is that we find other explanations more plausible than chance in those cases. If you flip a coin 100 times and get random-looking results: sure, those particular results are very improbable, but very improbable things happen all the time (as in fact you can demonstrate by flipping a coin 100 times). What you should generally be looking at is not probabilities but odds. That random-looking sequence is neither much more nor much less likely than any other random-looking sequence of 100 coin-flips, so the fact that it's improbable doesn't give you reason to disbelieve it -- you don't have a better rival hypothesis. But if you flip all heads, suddenly there are higher-probability alternatives. Not because all-heads is especially unlikely by chance, but because it's especially likely by not-chance. Maybe the coin is double-headed. Maybe it's weighted in some clever way[1]. Maybe you're hallucinating or dreaming. Maybe some god is having a laugh. All these things are (so at least it seems) much more likely to produce all-heads than a random-looking sequence.
[1] I think I recall seeing an analysis somewhere that found that actually weighting a coin can't bias its results much.
But if you flip all heads, suddenly there are higher-probability alternatives. Not because all-heads is especially unlikely by chance, but because it's especially likely by not-chance. Maybe the coin is double-headed. Maybe it's weighted in some clever way[1]. Maybe you're hallucinating or dreaming. Maybe some god is having a laugh. All these things are (so at least it seems) much more likely to produce all-heads than a random-looking sequence.
Which is, I think, what is interesting about this: All-heads is no more improbable than any other random sequen...
Alice: "I just flipped a coin [large number] times. Here's the sequence I got:
(Alice presents her sequence.)
Bob: No, you didn't. The probability of having gotten that particular sequence is 1/2^[large number]. Which is basically impossible. I don't believe you.
Alice: But I had to get some sequence or other. You'd make the same claim regardless of what sequence I showed you.
Bob: True. But am I really supposed to believe you that a 1/2^[large number] event happened, just because you tell me it did, or because you showed me a video of it happening, or even if I watched it happen with my own eyes? My observations are always fallible, and if you make an event improbable enough, why shouldn't I be skeptical even if I think I observed it?
Alice: Someone usually wins the lottery. Should the person who finds out that their ticket had the winning numbers believe the opposite, because winning is so improbable?
Bob: What's the difference between finding out you've won the lottery and finding out that your neighbor is a 500 year old vampire, or that your house is haunted by real ghosts? All of these events are extremely improbable given what we know of the world.
Alice: There's improbable, and then there's impossible. 500 year old vampires and ghosts don't exist.
Bob: As far as you know. And I bet more people claim to have seen ghosts than have won more than 100 million dollars in the lottery.
Alice: I still think there's something wrong with your reasoning here.