They still made a lot of assumptions. As shown in the video, they use distant stellar bodies to determine the distance of other stellar bodies. They began without proper angles to begin with. Even the man in the video admits that they didn't know trig back when they "figured this out."
Hence they used a lot of guesses, assumptions, and circular reasoning. They could not know the distance to the moon without knowing the diameter, and they could not know the diameter without knowing the distance. And they could not know the distance to the sun by that method without knowing the correct distance to the moon. And without trig., which they didn't have when they "figured this out," the whole thing is useless.
Using the method of light beams through clouds would be a sure way to accurately figure out the distance of the sun, provided you have accurate markers to where the sun is hitting on the ground, and you are on the equator during the summer in that location. Another method would be to take a measuring tape to the sun, lol. Or maybe if they ever do get past the Van Allen Belts they will read their odometer.