The Gelfand duality for pro-$C^*$-algebras

The answer is No. Rougly, because it is not a good idea to look at continuous $\mathbb{C}$ valued function on a space which is not completely Haussdorff as completely haussdorf is exactly the hypothesis that says "your space can be understood by looing at function over it"... I really don't think you can obtain something different from Philips result by considering the exact same functor.

More precisely: Take any example of a space $X$ which weakly Hausdorff CG, but non completely Hausdorff.

then there is going to be a pair of points $x,y \in X$ such that for any continuous function $f:X \rightarrow \mathbb{C}$. $f(x) = f(y)$

Hence the two functions :$x,y: \{ *\} \rightrightarrows X$ are going to have the same image by your functor which is hence not faithfull, and hence not an equivalence of category.

Edit : This answer the edited version of the question.

If you start from a weakly Haudorf CG space $X$, then you can consider the equivalence relation on $X$ defined by $x \sim y$ if for all continuous $\mathbb{C}$ valued functions $f$ on $X$, $f(x)=f(y)$. Let $Y$ be the quotient of $X$ by this relation. By construction $Y$ is CG (it is the quotient of a CG space), any functions from $X$ to $\mathbb{C}$ is compatible with the equivalence relation hence defines a function from $Y$ to $\mathbb{C}$, hence $Y$ is completely Hausdroff and $C(Y) = C(X)$. So the image of your functor is the same as the image of the functor of Philips when restricted to CG spaces, which as you mentioned in your question is apparently not essentially surjective.

One can also probably prove it is not full by constructing an example where there won't be so many interesting maps from the quotient $Y$ to the initial space $X$ while if the functor was full then the isomorphisms map from $C(X)$ to $C(Y)$ should be represented by a map from $Y$ to $X$...

The answer is yes, provided you change the category on the topological side slightly to: compactly generated functionally Hausdorff topological spaces with a distinguished family of compact sets; with continuous maps that preserve the distinguished family. See Theorem 6 of

Michael Forger, Daniel V. Paulino, Locally $C^*$ Algebras, $C^*$ Bundles and Noncommutative Spaces, arXiv:1307.4458v1