Prove that the electric field produce by a punctual charge is isotropic and radial

I would like to prove mathematically that the electric field produced by a punctual charge is isotropic and radial, i.e.

$$vec{E}(r,phi,theta)=E(r)vec{e}_rtag{1}$$

I think that this statement is visually understandable by invoking the idea that a sphere would produce a spherical electric field, but it seems hard to show it mathematically.

I know that the the charge distribution of a punctual charge located in $vec{x}=vec{x_0}$ is expressed as $rho(vec{x})=Qdelta(vec{x}-vec{x}_0)$. And from Gauss law, one would get:

$$iint_{mathcal{S}}left(vec{E}cdotvec{n}right),dS=frac{Q}{varepsilon_{0}}tag{2}$$

where $mathcal{S}$ is a closed surface which interior contains $vec{x}_0$. I don’t know how to proceed after this step. Maybe different kinds of surfaces $mathcal{S}$ should be tried. What would you do to prove the statement?