Chrome is never going to get a dedicated search bar, the address bar in Chrome is designed to get you where you need to go. If you know the direct url that's awesome; but if you don't that's no problem as a Google search will take you where you need to go.
I feel this is much better than the Safari/Firefox way of handling things, where you essentially switch between two modes (usually by tabing) of either trying to find something or trying to hit a direct URL. Users don't want to operate in modes, they just want to find things.
I actually disable the google bar in my Firefox because the awesomebar takes care of whatever I need. It's pretty much the same as google's search bar. I'm not sure why anyone feels the need to use the separate bar.
The reason there are two seperate bars is the privacy implication of having autocomplete from a search engine for every url you type in. If I want to go to example.com, that'll get sent to Google for autocomplete if I type it on the Chrome address bar or the Firefox search bar. Depending on what example.com is, I may not want Google knowing I went there. On the other hand, if I type it on the Firefox address bar, it doesn't get sent for search autocomplete (just history, bookmarks etc.), so Google doesn't know which site I'm going to.
That's the official purpose of having two bars anyway, but I can't find the article where I remember this being stated.
Yes, but then you miss out on auto-completion for searches. Hence, why I prefer the separate bars. If I know the URL or title of the page (for FF's Awesomebar), cmd+L (URL bar). If I want to search, cmd+K (search box).
It's all personal preference, and there's no universally correct answer.
I feel this is much better than the Safari/Firefox way of handling things, where you essentially switch between two modes (usually by tabing) of either trying to find something or trying to hit a direct URL. Users don't want to operate in modes, they just want to find things.