If we want to educate flat-earthers, we should do it with facts and strong reason, not with somthing that is easily picked apart.
Actually, I only partially agree.
If facts and strong reason were adequate, then there would be no need to argue with them. Even the "poorly executed" argument presented, is perfectly adequate to demonstrate that the Earth is round, whatever flaws it might have.
Most conspiracy theories have, at their heart, a combination of ignorance and mental illness. It might not be extreme mental illness. It might not even be anything that ought to be called mental illness at all. However, it has to be, at the very least, such a strong, unshakable, belief that one can totally ignore all the evidence put forward.
The flat Earth belief is one of the most extreme forms. In order to believe in a flat Earth, you have to be totally kooky. Rational argument isn't going to work in a case like that. Somehow, you have to get at the underlying source of the erroneous belief, and attack that.
I think the approach that I would take is to find some sort of trusted figure in their life, and show that they are actively complicit in the flat Earth conspiracy. It might be a preacher that they watch on TV, and find a clip of him talking about his sermon being broadcast by satellite. They might be sports fans. Point out that when the Yankees are playing on the west coast, the live coverage shows that it is still daylight when the sun has set in New York. (I don't know how or if a flat-earther would explain time zones.)
As for any argument that was based on Eratosthenes or Carl Sagan? A pagan and an atheist? No way will such an argument win, regardless of any existence of minor flaws in the argument.