The factually incorrect part is the hidden premise that "convinced that X has never happened" is equivalent to "assigns P(x) = 0". In fact, it is equivalent to "assigns P(x) = epsilon", where the number of possible sequences is several orders of magnitude greater than 1/epsilon.

It basically comes down to saying something like this:

1. 1/x ≈ 0 for very large x

2. x0 = 0 for all x

3. So x1/x ≈ 0 for very large x

4. But x/x = 1

Therefore

1 ≈ 0

This reasoning is correct, but the error tucked away in that last "≈" is exactly 1.

The fact that if you don't chose what you are looking for, any sequence you come up with is rare, isn't news. Since they specified a sequence, all heads, you can say it has probably never happened, even while acknowledging that a minute fraction of the various possible (but individually just as rare) sequences have happened.

The odds are 1.3×10^-23 times the number of times anyone has attempted it, so there's no reason to believe that it has ever happened.

No, the sense of the comment is completely wrong-headed with respect to probability. The probability that a sequence occurred is 1. The probability that the 10-10-10.... sequence mentioned has occurred is indeed as close to zero as the 76 heads.