They don’t need to limit themselves to HDMI power. They could do like the Chromecast and Fire stick do, and have an external power source for 7.5W or whatever over USB. At that power level, I’m sure Apple could develop or repurpose their existing silicon lineup to be able to make a passively cooled stick design for under $100.
How do you wake a Mac Mini? Is it enough to just press a keyboard button? If so, does the keyboard have to be wired, or does Bluetooth work?