This proofs that all charts you see are Mercator projections. The great circle on a standard chart will become a straight line on the rotational elepsoid (Earth). That's why the apparent longer line is in fact the shorter way. That's the proof you are looking for.
Discussion
Its only proof of a simple mathematical model and proofs nothing about reality. If the precondition is already the assumption that the earth is a sphere, of course it will show you exactly that. What if all maps are wrong in the first place?
I sailed myself around the world several times. It's a ball my friend. 👋