> But imagine this was a domain you weren't familiar with, you didn't know that pi != 4, you didn't know that the proof was false going into it. Could you have come up with a list of problems so quickly?

No, but if I didn't know anything about the domain, literally any proof (correct or incorrect) would seem fine. But then it's not really "proving" anything. Knowing enough for the proof to make sense but still unconditionally accepting assertions like "if you fold the corners an infinite number of times, it makes a circle" strikes me as odd.

> I'm not sure that problems 1 and 2 are actually genuine problems with the proof. You can approximate the length of a curve with straight lines by making them successively smaller.

But that's not what's happening here: the lines are straight, but you'd approximate the length of the curve with the hypotenuses, not the legs of the folds. Surely as you repeat this process you wouldn't think "wow, the circumference of this circle is actually equal to the perimeter of the original square." You'd have to disbelieve your own eyes and intuition and knowledge of circles to accept that this is true and hopefully you'd think "maybe I'm doing this wrong."

That's not to say 1 and 2 alone prove the visual proof incorrect, but they demonstrate that it is doing something wrong. Proofs that are correct don't have inconsistencies.