it's an 8BitDo product in the Framework store that wasn't designed or manufactured by Framework

Right, so it's the same problems as any other third party gamepad.

You can see the frustration in this LinusTechTips DIY build your own steam machine video: https://youtu.be/2psXxetNpoo?t=1250

You can see he's actually using a 8bitDo controller like you shared, but it doesn't have the firmware to talk directly to his computer, which then needs to have the correct CEC codes for HDMI to tell the different TVs to turn on/wake up/turn off.

So to make it a media center / steam machine you need to manufacture + firmware both the controller and the PC, which is the GH ticket above (i think!). But since they didn't make the controller, it would then be on users for each third party to figure out how to connect to whatever Framework exposes. Overall, just much better to make and ship the controllers & computer together, which is why Steam is going to do so well.

edit: Half that was directed towards the person who shared the controller, ;)