That's cool and all, but not sure if that counts as a thing we built as much as a thing we drew.

Exactly, yes! The LHC is so much more (larger isn't the right word, maybe massive?). If it was on the surface instead of being buried, and the earth was perfectly spherical, you wouldn't be able to see it standing in the middle of it, because the ring would be on the other side of the horizon all around you.