WANT greater depth of field. You want LESS.
That's what the non-photographer public senses when they talk about the difference between "professional photos" and "snapshots."
In a snapshot (small camera), everything in the picture is in sharp focus, which makes the photo about the "scene" and distracts eyes from any one particular subject.
Shooting at f/2 on a tiny sensor, you get only snapshots.
Shooting at f/2 on a DSLR, only the subject (the person, the face, the rock feature, whatever) is in focus, and everything else is slightly blurred, which brings attention to the subject of the image, and at the same time blurs out distracting, unimportant details in the background.
Here's a good example from Google Images: http://ns12.sovdns.com/~nich61...
On a small camera or a smartphone, only the photo on the left is possible. In fact, on the smallest phones/cameras, you won't even get that much blur in the background; nearly everything can be razor sharp.
Generally, that's not good for subject work—only for scene work.