Nowhere? I pretty much stopped reading there. The author is obviously misinformed.
Maybe it came out of a few simple observations, and a little out "of the box" thinking? The observations: The sun is round, the moon is round, the stars are round, the planets are round. The moon revolves around the earth, the sun (seems to) revolve around the earth, the planets (seem to) revolve around the earth, the stars (seem to) revolve around the earth. The out "of the box" thinking: maybe the earth is round, too? Maybe the celestial bodies are a long way away, and revolve around something else?
I’ve not been impressed over the years with the level of discussion about the Antikythera mechanism (not that anyone asked me of course), but there’s nothing in the working replica of it to indicate that it’s based on anything but a fairly short period of observations done more or less locally (Aegean), and given that it’s a maritime culture, probably wasn’t built for ceremonial purposes, but as an aid to figure out the tides.
There’s a topic on FR about recent finds on the thing, including Greek place names, and the claim that it was used to figure out when the Olympics were going to take place. I’m pretty sure though that the Greeks knew how to count to four, and wouldn’t need a machine to tell them when to hold the Olympics.
The problem with the thing is that it’s unique; the exact date of its construction has been debated over the years, but the range of years during which its mechanism would have been accurate shows when it was made (probably on the early end of that range).