Anyone having trouble with unintended diagonal inputs on the matching 8bitdo controllers, it’s caused by the pivot nub under the D-Pad being slightly too small. It’s the same problem the NES Classic Edition had, so the fix is the same too:
My twin brother clearly knows how to stress a D-pad with Panel de Pon / Tetris Attack.

It’s still not perfect but it’s MUCH closer to an original SNES controller and most will consider this a total fix.
Remember:
Nintendo couldn’t get this right with the NES Classic Edition and the SNES ASCII Pad had the same issue way back in the day, so let’s not act like 8bitdo needs to be held to a higher standard than Nintendo themselves (or their licensees). It’s a nit-pick-fix.
Another thing I haven’t thought about to much is that a GBA player with HDMI is bound tk be made as well.
Already exists. There are multiple GameCube HDMI solutions now that convert directly from digital RGB (no analog). Plug and play HDMI cables/dongles exist for the DOL-001 GameCubes but DOL-101 requires an internal modification.
Unfortunately, the GB Player Start Up disc is not adequate so you will need a way to boot GB Interface instead. You can do that using hard or soft-modded ‘cube or you can just buy a Datel SD Media Launcher.
Yes, you can soft-mod a ‘cube using an Action Replay if you have some way to write to a memory card, like a hacked Wii or another GC. Some setups require a 3rd party memory card to install the loader due to protections.