Let $\Omega$ be a sufficiently regular domain, for example $\Omega=B(0,r)$, $r>0$, and $m(x)=\textrm{dist}(x,\partial \Omega)$ be the distance to boundary function.

Suppose $\mathcal{F}$ is a given family of smooth functions, which is dense in $C^m(\bar\Omega)$ (with corresponding $m$-th order sup-norms) for arbitrary $m$. Is it possible to explicitly construct another family $\mathcal{G}$ of functions based on $\mathcal{F}$ and the function $m(x)=\textrm{dist}(x,\partial \Omega)$, such that $\mathcal{G}$ is dense in $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ (i.e., any function $g\in \mathcal{G}$ satisfies that $g|_{\partial \Omega}=0$)?

A natural candidate is the set $$ \mathcal{G}=\{ mf\mid f\in \mathcal{F}\}, $$ where I directly multiply the function $f$ by the distance function. Since $\Omega$ has a sufficient regular boundary, the function $m$ is sufficiently smooth, hence for any $f\in C^{2}(\bar{\Omega})$, $mf\in C^2(\bar{\Omega})\cap C_0(\bar{\Omega})$. I also know $C^{2}(\bar{\Omega})\cap C_0(\bar{\Omega})$ is dense in $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ (in $W^{2,2}$ norm). But I was not able to proceed further to show $\mathcal{G}$ is dense in $C^2(\bar{\Omega})\cap C_0(\bar{\Omega})$.