I find it fallacious to see this as a criticism of p-values. If I wanted to identify whether a particular coin was the [most] biased out of 10^1000, flipping o

I find it fallacious to see this as a criticism of p-values. If I wanted to identify whether a particular coin was the [most] biased out of 10^1000, flipping one coin 10 times isn't a reasonable setup for the experiment. And, my conclusion to your hypothetical experiment would be: the person who told me that there was a 99.9...% chance this coin was fair was likely incorrect about that.