Update: Fixed the title of "Pikmin". Accidentally wrote "Pikman".
Considering the recent Nintendo license requirements, I expect that their demonstrative YouTube videos will have a difficult time staying online. Regardless, if you are in a jurisdiction where this is legal, it is now possible to play some Gamecube-era games at 60 FPS (as well as 1080p) with an emulator PC.
The blog post at the Dolphin Emulator's website goes into the “hack” in detail. The main problem is that these games are tied to specific framerates, which will cause problems with sound processing and other, time-sensitive bits of code. I have actually been told that one of the most difficult aspects of bringing a console game to the PC (or restoring an old PC game) is touching the timing code. It will break things all over. For Super Mario Sunshine, this also involves patching the game such that certain parts are still ticked at 30 FPS, despite the render occurring at twice that rate.
Also interesting is that some games, like Super Smash Bros. Melee, did not require a game-side patch. Why? Because they apparently include a slow-motion setting by default, which was enabled and then promptly sped up to real time, resulting in a higher frame rate at normal speed. The PC is nothing if not interesting.