I've seen that explained. I wonder if TheGnome will explain it in the same way.
Erastothenes did not prove a spherical earth. This fact was known long before. It was inferred e.g. by the nature of the earth's shadow on the moon during an eclipse.
A spherical earth was his starting point. What he did was determine its size. For that he used a rather simple geometrical argument. In it he used three facts that were well known by then.
First the earth is spherical. That enabled him to use the circle with its well understood properties.
Second the sunrays arrive on earth almost perfectly parallel. That made the math quite simple.
Third the earth is rotating on its axis and he had to know its orientation. That enabled him to choose 2 locations with the same longitude and to time the measurement.
That's my take on what he did. It was an impressive achievement and I never doubted it in this whole thread. Nor did I in any way doubt what science does know about the solar system.
But when I look at the presentation under discussion (that by the way doesn't outline the work of Erastothenes), I don't find the slide to mention that the sun is not small but far away

. And without that the logic is just not sound.
I get it that today everybody that is not a flat-earther knows this very well by his teens at the latest and he fills in the missing bits with ease. But then this presentation doesn't tell him much new anyway.
But to teach the intelligent uninformed, or to convince a flat-earther even, such details are important to make a valid argument.
Now I take it that the presentation was not meant to be standalone so I give it some slack. But even so, it outlines an argument for a curved earth and leaves out a basic ingredient which would have needed just one aditional slide! Without it the conclusion is just not valid.
I can only assume that this was an oversight by Sagan.