Even if the pure slingshot without using the engines is debatable in this scenario you could definitely use the Oberth effect to dramatically reduce the amount of fuel needed to launch rockets into space. The rocket could accelerate from the surface toward the center of the earth without burning any fuel at all and then fire its engines when it reached the center. It could then achieve the escape velocity using a lot less fuel than would require in a conventional launch.
It is not debatable. You need to be in a different orbit than the body you get the slingshot from. Period.
You make a good point about the Oberth effect, but I'm not sure how it would work in this situation. One problem is that the math no longer behaves when you're "orbiting" inside the body. For starters, you no longer have a constant gravitational parameter (effectively mu = 0 at r = 0), so you can forget about using vis viva.