While I agree with you that accidental discovery is sometimes valuable, it seems like introducing you to things you weren't necessarily looking for was an unintended side effect of how the web originally worked. The goal was to help you find what you were looking for, but because it didn't always work completely, you sometimes ended up finding what you were not looking for.
What's interesting is that while Google is narrowing down on what you are specifically looking for in search, it is simultaneously introducing a product that is designed to suggest things you have not explicitly asked for: Google Now. It will be interesting to see if they can provide the discovery functionality that you want with a product that is specifically designed to perform that function, rather than one that does so unintentionally.
It's not accidental discovery per se that I miss. I'm more worried about the subtle moral, social and political implications that the loss of impersonal information might bring about. Will Google rank negative information about my favorite Congressional candidate lower in its search results, because I often email my friends with positive information about him? How long before a Wikipedia article about some scientific fact gets ranked below misinformation that an annoying relative has emailed me about 20 times last year? I'm a firm believer in the idea that the best way to make most people open-minded is to force them to confront diversity in their daily lives. But the more you are surrounded with personalized stuff that appeal to your appetites, the less likely you are to confront the Other. Of course I don't want Google forcing anyone to read anything, but I don't want them to discourage people from reading certain things, either. I want my information sorted by relevance, not anyone's political biases. Accidental discovery was not an unintended side effect of the design of the web. It is a natural consequence of the fact that facts are impersonal.
I'm also not sure whether something like Google Now could fill the role of completely unpersonalized, "objective" information. It is still personalized in the sense that the algorithm is tainted by my current preferences, so it might confirm my biases in more subtle ways. In fact, Google has every incentive to show me results that please my palate as much as possible. "Worried about Obamacare? Why don't you check out this fundamentalist's blog?"
What's interesting is that while Google is narrowing down on what you are specifically looking for in search, it is simultaneously introducing a product that is designed to suggest things you have not explicitly asked for: Google Now. It will be interesting to see if they can provide the discovery functionality that you want with a product that is specifically designed to perform that function, rather than one that does so unintentionally.