Epicycles are simply "a small circle whose centre moves round the circumference of a larger one".
Of course astronomy still uses epicycles because that's how the universe works. The earth orbits around the sun, and the moon orbits around the earth.
On a larger scale the sun orbits the galactic centre.
The mistake in astronomy pre-Copernicas was to think that everything revolved around the earth. When planets were observed in retrograde motion epicycles were added to attempt to correct that. It's not that epicycles themselves are wrong, it's just that they chose the wrong ones. And they did that because:
Geocentrism lasted so long because priests and philosophers decided that the Earth’s placement in time and space had implications for its spiritual importance. That made accepting heliocentrism anathema on a deeper level than science, which is the reason so many smart people toiled to preserve something whose falsity was increasingly obvious for centuries
That's a quote from one of your articles above, Tom.
You say "The models of Astronomy are still in the stone age", but they demonstrably work. They're not perfect by any means, but they've got us to the moon, they've got probes to all the other planets. They enable us to make predictions on all kinds of astronomical events. There's a lunar eclipse on Sunday. You can find out where and when to see it at your location, you can then make an observation and compare that to the prediction.
Now all you have to do is use your flat earth model to make your own prediction, see how well it compares with the predictions made using mainstream astronomy and you can see how well it matches up with the observations. Then we can talk about which model better matches the reality. Good luck! You've got a few days before Sunday so you can post your predictions here if you like, showing your workings.