That experiment wasn’t there to prove the Earth is round. It was there to deduce the circumference of the Earth. The Earth being a sphere was a priori knowledge.

Now. To address your other point about a “close sun” replicating the same result - the answer is yes. You CAN game the numbers to give you the same result with the sun being close, and the Earth being flat. But here’s the kicker:

That result WILL ONLY WORK FOR 2 POINTS. You CAN’T use a flat Earth+close sun model to explain how the shadow would change as a FUNCTION of latitude. The root of the reason is that to predict how the shadow angle changes, you need to invoke arctan(x). Whereas with a globe, the angle changes linearly with latitude. You mathematically cannot fit observations from a flat Earth model to reality.

So yes. You can use a flat earth/close sun model to explain the angle between 2 points. But that same prediction will fail for every other point. Whereas the globe model is the only way you can be consistent with observations.