...at least, that is not what this amateur balloon flight shows:
I have to agree with Auguste Piccard, "a flat disc with up-turned edges".
How could you know that you "agree with Auguste Piccard", when we're you at 50,000 feet looking through the tiny port at the horizon?
You claim to be a "physical observer" now you are claiming to "agree" with something you can never have observed.
102 miles in the air{2:24:00}, and the flat level horizon stays at eye level.
Try again
Mr Physical Eye-Wide-Shut Observer, at 144 min the balloon is at 40,639 feet, or about 7.7 miles, not 102 miles in the air.
Not only that, however could you tell that "the flat level horizon stays at eye level"?
The balloon is wobbling and rotating and there is no horizontal reference.
From 40,639 feet the "dip angle to the horizon" is only about 3.6° and there is no way that anyone could detect that from those photos.
By the way, the highest altitude of that balloon was a little over 117,550 feet or 22.26 miles, at 124:17.
During 2002 an ultra-thin-film balloon named BU60-1 made of polyethylene film 3.4 µm thick with a volume of 60,000 m3 was launched from Sanriku Balloon Center at Ofunato City, Iwate in Japan at 6:35 on May 23, 2002. The balloon ascended at a speed of 260 m per minute and successfully reached the altitude of 53.0 km (173,900 ft), breaking the previous world record set during 1972.
That is under 33 miles and is that js the current record.
At 72 miles{49:50}, the camera was dang near eye level with the sun. Even I went, woe!
Stop wasting everybody's time with meaningless incorrect rubbish!
I do not mean that the balloon flight is not a great achievement, just the silly conclusions drawn from it.
But, I've been on eyelevel with the sun too! See here, where the
sun is obviously is obviously half way below the horizon! 
What height did you say the sun was again?

Yes, the sun really does sink to below our eye-level, from our perspective.
PS Your original error was taking the horizontal distance the balloon moved for the, clearly labelled, altitude.
If you had a little more actually knowledge of these things, you would realise something was wrong, but
You are trying to decide the shape and motion of the earth, when you obviously have minimal knowledge on the matter.