You chortle about the buildings one, and yes, for structures like houses there is literally no perceptible difference, but certain large bridges around the world have, so their architects claim, been built with pillars wider apart at the top than at the base - on the largest bridges in the world, the difference actually amounts to several inches. However, when the bridges are actually constructed, they are always built parallel. You can consult low-level bridge engineers on this, or go and measure them for yourself.
Anyway, I think the most serious problem with the Copernican model is that a lunar eclipse should not be able to occur during broad daylight, yet many have done. The RE model is elegant, certainly, but not quite elegant enough to explain that particular phenomenon to any degree of satisfaction. It just doesn't fit. It can't.