I've been running with a GTX 1060 (3 GB model) for the past three years. It works great and can handle 4K with many games, if you don't push too many other enhancements on it.
However, I'm going to suggest something slightly controversial, which is that there really isn't any point in running GameCube or Wii games at 4K resolution. They weren't designed for it, and even with custom texture packs I see little benefit of 4K over 1080p in those games on a 65" 4K display. Standing unnaturally close to the TV, I thought 1440p looked marginally better than 1080p in Skyward Sword with custom textures, but I actually thought the GPU resources were better spent on SSAA anti-aliasing than on additional resolution. So now I run Skyward Sword at 1080p with anti-aliasing and anisotropic filtering, and feel like that's the best combo.
You (MonicaPuig) mention wanting to play GameCube games, which see even less benefit from 4K, IMO. Try running your games at 1080p with anti-aliasing and anisotropic filtering and see what you think.
However, I'm going to suggest something slightly controversial, which is that there really isn't any point in running GameCube or Wii games at 4K resolution. They weren't designed for it, and even with custom texture packs I see little benefit of 4K over 1080p in those games on a 65" 4K display. Standing unnaturally close to the TV, I thought 1440p looked marginally better than 1080p in Skyward Sword with custom textures, but I actually thought the GPU resources were better spent on SSAA anti-aliasing than on additional resolution. So now I run Skyward Sword at 1080p with anti-aliasing and anisotropic filtering, and feel like that's the best combo.
You (MonicaPuig) mention wanting to play GameCube games, which see even less benefit from 4K, IMO. Try running your games at 1080p with anti-aliasing and anisotropic filtering and see what you think.