So now I'm thinking - is the solution to support multiple options or to own all pieces of functionality?
I'm honestly not sure I can answer that. I could argue either direction. I think it boils down to how important the functionality is to your application. It seems like risky business to rely on a single third party for mission critical components of your operation. Also the assumption that ALL users you'll encounter will have an Amazon account is... optimistic. At least a payment service like PayPal offers a login or a blind CC processing option using the cards details. This lowers the bar, I've had a PayPal account for almost 10 years, I've processed multiple payments with it PayPal. In the past 4 years I haven't logged into my account to do so, I'll type in my actual card information rather than deal with remembering my password, recovering my password, updating my stored card information, etc.
So why don't you just sign up with a different email like Amazon suggests in their email?
Well for one, I shouldn't need to. My account is healthy and in good standing across every other line of Amazon products. The ONLY service I use it for is currently KickStarter. To top it all off, I really don't care all that much. I just threw this post together after thinking about what adopting third party services for core features of my own future products and what unseen pitfalls that might entail.