I was thinking about it and although i would like it for its simplicity there is one problem with this approach: How will a new user know that this is also a search box?
The way chrome does it is that when you first open it, the focus is on the address bar and a search icon suggests that you can input a search query.
With MusicBee it's different as it displays the current location in the address bar when you open it. So with address bar already occupied how do you let the user know he can type in a search query?
I'm not saying that this can't be done, it's just that the way you want it to be wouldn't be very intuitive. I'm sure you can come up with something better.