Nvidia got the bid for Switch when they were basically dumping those unwanted Tegra to Nintendo for an incredibly low price.
Xbox and Playstation dont earn AMD much profits at all. AMD had this Custom Processor segment to barely keep them surviving, people may forget AMD was only worth ~$3B market cap in 2016. They are now at ~$250B.
On the subject of software compatibility though, one thing I got it wrong was my prediction of having AAA titles on Xbox and PS would have helped AMD's market share on PC, given those titles are already optimised on Xbox and PS anyway. That didn't happen at all. And Nvidia continue to dominate.