Yeah, I don't think arbitrary code execution counts as the game itself being "Turing complete".
On the other hand, what is "Super Mario World" if not the game as it runs on the SNES?
Yeah but it's a different kind of fuckup, designing a system with a vulnerability that lets you "escape" into the underlying machine, and designing a system that, per-se, turns out to be Turing-complete.
Same with pokemon, they just found a bug that lets them write to arbitrary memory.