What core games, using modern programming techniques, couldn't be made on Gen 6?

[novasonic]

Gamecube was more high end graphics-wise for the time but whichever.

The answer to your question is none. The only thing that has advanced significantly since then is graphical prowess. They all had built-in or optional modems for online. They all could have had motion control add-ons. Number of controllers have changed, but controller design has not. The only thing that makes a new console interesting now is its exclusives.