The 'Android has problems' narrative is very clearly established (as established, of course, as the 'Android is selling in vast numbers' narrative). There's the fragmentation by OS version, the fragmentation by hardware, the consistently lower engagement and monetisation, and of course the financial weakness of all the branded OEMs except Samsung, which has an equally unhealthy (for Google) 45-50% or so of Android device sales.
However, it is far from clear to me that problems for Android developers and OEMs are the same as problems for Google.
As I see it, Google really has three strategies for Android, as it has developed over time.
First, there was the publicly stated objective; to make sure that Google was not somehow shut out from the mobile internet by a dominant OS provider that chose to exclude it. Originally (i.e. pre-iPhone) this fear was directed at Microsoft; subsequently it may also have been consciously directed at Apple.
Android has been a complete and unambiguous success on this front: it is extremely hard to see how any one company could now control the mobile OS market at all, let alone try to exclude Google. (The exception, of course, is China, which I will return to.)
Second, whether deliberate or not, Android has had the effect of hugely increasing the number of people with access to the mobile internet. Just as Wintel 25 years ago powered an army of cheap PC 'clone' makers churning out tens of millions of cheap commodity PCs, Android and a small group of mobile chip companies (mainly Qualcomm, EMP, Mediatek, Spreadtrum) have enabled a flood of cheap commodity smartphones and tablets. Do a search on Alibaba for 'Android 2.3' to see what I mean - the entry price for an Android smartphone is now $45 wholesale, with wifi tablets not much more and a vast range of other devices (ersatz netbooks, in-car PCs and DVD players, set-top-boxes and lots else besides) following on behind.
These devices in turn are percolating out to emerging markets around the world and to prepay customers everywhere, quite apart from the relatively small numbers of people in developed markets buying relatively high-end devices like Samsung's Galaxy S3 (which made up under 10% of Android device sales in 2012). All of those devices represent more people and more time online, and that means search, which means Google web search and revenue for Google.
On this front too Android has been a huge success, and many of the criticisms of Android as an ecosystem are of limited relevance to Google. A forked, fragmented, Android 2.2 phone in Jakarta with no Google services on it still accesses the web (if it has a data plan, of course) and drives search volumes, and it is still a great product for a labourer with $50 to spend. The decline of HTC matters little compared to that. Even the dominance of Android sales by Samsung, though arguably unhealthy, doesn't fundamentally threaten these strategic gains so long as the 'cheap Chinese' are providing a flood of alternatives.
Equally, iPhones, which have around 20% of the smartphone market, do almost all of their web search (outside China) on Google. Given the fact that $650 phones tend to be bought by higher-end consumers than $100 or $300 phones, it is quite possible that iPhones generate more advertising revenue for Google than all Android phones combined.
Then, however, we come to the third strategy.
An Android device, properly signed into a Google account and running all the Google Apps, generates an endless stream of little bits of 'signalling' information, way beyond what Google gets from a desktop search user even if they're using Chrome. It knows where you live and work, how you commute - and which phone numbers on web ads you dial. Unlike a web browser, you are probably always signed in to Google, so all of your interactions with Search, Maps and anything else can be linked together. (This, of course, is also the main purpose of Google Plus.)
Google Now is just one manifestation of this: doing useful things like telling you that your meeting will take 45 minutes to get to because there's heavy traffic, so you should leave now. But this value flows both ways: Google is giving you useful titbits, but it is also mining far more data than it would get from a PC user - data to improve search relevance and advertising relevance.
In other words, Android, like Plus, allows Google to tie searches and advertising to individual people and places. In the long term, the data that Google gets from Android users is probably just as important as Pagerank in understanding intent and relevance in search.
Hence, the real structural benefit to Google from Android now comes from the understanding it gives of actual users, and the threat comes from devices that do not provide this data - even if, like the iPhone, they do provide plenty of search traffic.
Obviously, Google's access to this data on non-Android platforms is pretty partial, but the problem also applies to significant parts of the Android base. In China the problem is near-absolute. Google services are mostly blocked anyway, almost all Android phones ship with few or no Google services installed (i.e. they are based on the open source AOSP version of Android) and Google Play, where it is installed, appears to have a very low share of actual app installs. Hence, Google gets close to zero practical benefit from the explosive growth of Android there. There is a similar issue in other emerging markets - a significant portion of those $45 handsets skimp on Google apps just as they skimp on IMEI numbers.
The clearest expression of this is in tablets. Google gets no data from a Kindle Fire, only web search traffic. More importantly, at scale, it gets no data from many of the generic Chinese Android tablets that are starting to well out of China at $100 or less. These devices are like dark matter: everyone suspects there are a lot of them, but no-one quite knows (publicly) just how many. I've seen credible people claim it could be well over a hundred million units in 2013. Possibly much more. How many of these devices will have Google Play? How many users will install Google Maps? How many will come with a third-party web browser (from Tencent, say), one or two of the dozens of major Android app stores operating in China, or Amazon's app store?
Someone (sadly, I forget who*) described Android to me as an unguided missile: very powerful but spiralling semi-randomly with no clarity on where it would land. There is the fragmentation issue, and the weakness of most of the OEMs. There is the threat of Amazon or Samsung forking the platform. But there is also the threat that an increasing number of Android devices might have no more connection to Google than does an iPhone.
To put that another way, Google's penetration of Android is as important as Android's penetration of the handset market.
*Update: it was originally a comment on Asymco - thanks Horace.