Chrome OS and Android: why have both?

I'm seeing this question asked in a lot of the recent threads on the Chromebook Pixel, and I figured a dedicated discussion might be in order.

Why is Google throwing its weight behind two separate operating systems when it has no legacy- or inertia-related reason for doing so? And no, I don't think the answer is "Google is super-decentralized and everyone does what they want". A company like Google doesn't just accidentally spit out a $1300 flagship laptop (though it might on occasion accidentally spit out marginal products like the Nexus Q). The Chromebook Pixel is evidence that Google is committed to Chrome OS.

But why? Google is in a unique position vis à vis its principal competitors in that it got into the operating system game by way of its mobile OS, and not the other way around. It has no desktop OS baggage. Furthermore, its mobile OS is extremely popular, extremely flexible, and extremely well-designed. It doesn't need to develop a separate desktop OS in order to have a desktop OS - it could conceivably just take Android and apply a different, more desktop-appropriate UI (hell, it could just apply the same UI that Chrome OS currently uses) and various other customizations.

The advantages of doing so are obvious. It would save money. It would save time. It would enable the two teams, and therefore the two OSes-that-were, to have closer synergy with each other (this is what Microsoft is currently attempting, bit by bit; Google could do it far better). It would give the fledgling Chromebooks access to a wide variety of attractive Android system features (like notifications) and apps (though they would have to be aggressively curated, I think, to make sure that they were genuinely desktop-compatible apps).

What are the disadvantages? The reason why Apple doesn't merge Mac OS and iOS, besides inertia, is simply that there is a lot you can't do with iOS that you would need to be able to do on a Mac. But that doesn't apply here. Android is flexible - certainly moreso than Chrome OS, which is famously "just a browser". Google is in the bizarre situation of producing a desktop OS that has fewer capabilities than its mobile OS!

Is it because they're worried of larding Android up with too much desktop-specific code? I'm not a programmer, I don't know how big of a concern this is. But considering Chrome OS is basically just Chrome I don't see how this could really be the case. You might need to make a few changes beyond simply altering the UI, but surely not many, and many of them would probably be beneficial to mobile Android too.

Would Android not perform well scaled up to desktop sizes and desktop processor speeds? I don't know, but Chrome OS doesn't really perform that well as it is - battery life is consistently disappointing, for one, and the Pixel seems to have problems with touch scrolling.

So what is it? What's the answer? Why is Google committing itself to two different operating systems when they already have one that could fill both roles just fine?