I don't know if Amazon is the right investor for the brand. But as far as I can see, the problem with the XBox at the moment is, that Microsoft don't want to invest much into studios and risky games. The 1. party lineup has gone very small, and that is a huge disadvantage. They may make a small benefit with the Xbox division, but they are loosing market share.
So if they sell the Xbox division, it's not the point to whom. Amazon, Apple, Google, SEGA (lol), whoever gets in charge, needs to invest, take risks and challange Sony and Nintendo. This may not be the safe way. But it's the only way to get back to top position.