A PCB wouldn't burn out without also taking the cartridge port with it and completely showing failure burns on the board, that's assuming we are talking about an electrical problem. If it's melted plastic that's even more suspect, especially considering an OG GameBoy w/ Tetris can survive being bombed and charred. A leaking battery or capacitor would easily show corrosion or residue build up.
For all anyone knows they could have botched an attempted battery replacement while also melting the case while soldering. If you tested the game before shipping that's even less of a reason to refund. If you sold this through ebay they will eventually end up siding with the buyer, for that reason its a good idea to state in bold sold as is. To prevent someone from swapping boards it's also a good idea to use a marker that is only visible with under a black light to tag a board with something you can identify.
The picture is way too low res to be able to determine what's going on. Try getting the guy to take better pictures of all sides of the board with a camera that isn't a toaster smeared in Vaseline.