"Embrace, extend" was more effective (and terrifying to competition) when Microsoft was the juggernaut. But now Microsoft is actually behind Google, so there's little chance they can extend it any meaningful way that would give them leverage against Google. Google has equal engineering resource and huge head-start in install base and ecosystem.
So the real question is what could Microsoft leverage from AOSP? My instinct is they could probably build anything from AOSP into WinPhone far faster than they could utilize AOSP directly.
Your instincts are wrong, but I've run into this misconception a lot. Nokia made this error, and RIM/Blackberry are doing it all over again.
All they have to do to use AOSP is 1. start using it, 2. develop apps which are clients for their services, and 3. release a device with those clients preinstalled. There is no magic hidden step.
Everything from the accounts system, inter app communication, billing, push, is pluggable and they already have the servers for most of it setup, which is far and away the hardest part.
Why do you think apps wouldn't be compatible? It would be just like Amazon - yes, you'd need to run a new store (fairly trivial for an MS or Nokia type org to get going) and need a new billing API, but that's all you'd need to get going.
To put this in perspective moving Play Store builds to the Amazon store is generally a matter of replacing the billing and push APIs, if you've used them, which takes a few hours. (If you've used game services it might take longer, but again it's not actually hard). This is far easier than redeveloping your app completely for a different platform.
I've been involved with apps for which the move to Amazon was simply a 10 minute boot up on a Kindle Fire to establish it worked.
Nokia are bad at it, but you might recall the N-Gage that had a services layer that wasn't completely terrible. In Europe the operators usually ran this stuff, at least pre-Apple/Google, and there were a good few companies that existed by selling portals which were white labelled and branded as the operator stores.
These things certainly can be killed in bureaucracy, but the tech issues aren't too bad, especially if you're already sitting on a giant pile of servers and a billing solution for each country. At one point I knew one team running stores for about 20 operators (each country is essentially separate) with less than that in developers, and only a handful of sysadmins, if you can have such a thing.
I'm not sure what misconception your think I have, but it has nothing to do with how easy or hard AOSP is to use. It's the fact that Microsoft already has a good mobile phone OS tied into their ecosystem. What does AOSP bring to the table that is going to move the needle for them?
Even better, Microsoft already license the Exchange protocols to Google and Apple, and then they get to pay Microsoft and do the work of integrating their mail and calendaring software with Exchange.
So the real question is what could Microsoft leverage from AOSP? My instinct is they could probably build anything from AOSP into WinPhone far faster than they could utilize AOSP directly.