The Kindle Fire is actually quite compatible, though. The launcher/core experience is very different, but the app selection is basically identical. GMS has some nice APIs, but the truth is that most apps (by download count, or by revenue) either don't use them or can configure them out.
The kindle isn't successful because Amazon was able to eject compatibility concerns, it's successful because Amazon was (unlike basically all the other variant Android customizations) able to bring elements to the platform that customers actually wanted, and that these outweighed their desire for interoperability with the Google GMS ecosystem.
According to a recent email I got from Amazon App Distribution: "75% of the Android tablet apps tested by Amazon already work on Kindle Fire without any extra development."
The kindle isn't successful because Amazon was able to eject compatibility concerns, it's successful because Amazon was (unlike basically all the other variant Android customizations) able to bring elements to the platform that customers actually wanted, and that these outweighed their desire for interoperability with the Google GMS ecosystem.