But what about the named club sandwich, which 1) has a width to height ratio so unfavorable it needs toothpicks to hold it together, and 2) often has more bread than ingredients (at sub-par establishments), but sometimes far more ingredients than bread (leveraging the toothpicks).
Regardless of these facts and it not meeting your criteria, it is still both in common usage and by explicit name a sandwich!
It's not the thickness of the sandwich, it's the thickness of the individual things in the sandwich. Everything in a club sandwich is flat, there's just a lot of stuff.
(Sorry, can't resist keeping this going, this is some good discussion).
It's true I guess that the aspect ratio for most ingredients is 'flat', but many are thicker than the bread, and the whole sandwich is 'sideways' aspect ratio wise.
Regardless of these facts and it not meeting your criteria, it is still both in common usage and by explicit name a sandwich!
(Also, love it!)