Thank you! This was very helpful. Maybe I'm missing something here, but couldn't you apply the same argument for $f(x)^n$ without using the Hausdorff distance result to drop the power, and get that $$\int_{S^{n-1}} |f(x)^n - r_0| dx \leq \epsilon$$ ? This would give a constant independent of the dimension.

You are correct that the statement follows, but with an estimate of order of $\epsilon^{1/n}$ for the distance from the ball. I was hoping that one can get a better estimate without using Hausdorff distance.