It wouldn't.
It is a well-known fact from gravity, which also has a law $\nabla\cdot\vec F_g \propto \rho$ expressing a $1/r^2$ force, that the force of a spherical shell of mass is zero inside that shell, but outside it is $-\hat r G M m / r^2$ as if all of that mass were at the shell's center.
The same mathematics will mean that the $\vec E$ field within a spherical insulator with uniform charge density $\rho$ will be $$\vec E = \hat r \frac{Q_{\text{encl.}}}{4\pi\epsilon_0 r^2} = \hat r \frac{1}{4\pi\epsilon_0 r^2} \rho \cdot \frac{4\pi}{3}r^3 = \hat r \frac{\rho r}{3\epsilon_0}$$which is plainly not zero unless $\rho$ is.
Interestingly, you can kill these sorts of arguments by extending the radius of the sphere out to all space, at which point Gaussian arguments get a little hazier and there are symmetry reasons to think that there is no net force on you. However, Gauss still wins in this way: any little noise in that background distribution of charge gets amplified by the laws as time goes on, eventually leading to a very uneven distribution of charge, so it's like balancing a pen on its tip: the physics says it's possible but unstable.