It is related with the large-scale geology of the USA and Europe. The USA is rather normal in this regard, but the particular geological history of Europe allowed the formation of exceptionally deep caves there.
During Mesozoic, Europe was a closed ocean: the Tethys Ocean, with shallow carbonated platforms. The Alpine Orogeny formed the continent and the carbonaceous materials emerged creating mountains with large carbonaceous units. The meteorization and erosion of these carbonates created big karstic systems with deep caves.
Conversely, the USA has a lot of Paleozoic units where the carbonaceous materials are too compact to form deep caves. You may have shallow Mesozoic caves near the surface but Mesozoic units are thinner than some folded zones of Europe.