The construction you describe when $\mathscr{C}$ consists of all closed sets of $X$ is known as the Wallman compactification of $X$. I'll denote if $\omega(X)$. It is due to Wallman; Lattices and topological spaces, Ann. Math. 39 (1938) 112-126.

Of course some sort of techincal assumption is required.

Let $X$ be a $T_1$ space. Then $\omega(X)$ is a compact $T_1$ space containing $X$ as a dense subspace. Moreover it has the property that every continuous map $X\rightarrow K$ into a compact Hausdorff space $K$ extends over $\omega(X)$. The space $\omega(X)$ is Hausdorff if and only if $X$ is normal, and in this case $\omega(X)\cong\beta(X)$.

Shanin later generalised Wallman's construction; On special extensions of topological spaces, Dokl. SSSR 38 (1943) 6-9, On separation in topological spaces, Dokl. SSSR 38 (1943) 110-113, On the theory of bicompact extensions of topological spaces, Dokl. SSSR 38 (1943) 154-156. The compactifications that Shanin constructed allowed for the ultrafilters to come from more general lattices $\mathscr{L}$ of closed subsets of $X$. Of course at the expense of added assumptions: $\mathscr{L}$ is required to be a so-called $T_1$-base for the closed subsets of $X$. Denote by $\omega(X;\mathscr{L})$ the Wallman-Shanin compactification built using the $T_1$-base $\mathscr{L}$.

Here are some examples to convince you that these compactifications are interesting.

1. $X$ is locally compact $T_2$ and $\mathscr{L}$ consists of all $(i)$ compact subsets of $X$, and $(ii)$ all closed subsets $A\subseteq X$ for which there is a compact $K\subseteq X$ with $A\cup K=X$. Then $\omega(X;\mathscr{L})$ is the Alexandroff compactification of $X$. Here are some examples.

2. $X$ is Tychonoff and $\mathscr{L}=\mathscr{Z}(X)$ is the collection of zero sets. Then $\omega(X;\mathscr{L})\cong\beta(X)$, as you have recognised.

3. $X$ is rim-compact $T_2$ and $\mathscr{L}$ is the set of all finite intersections of regularly closed sets with compact boundaries. Then $\omega(X;\mathscr{L})=\mathfrak{f}(X)$ is the Freudenthal compactification of $X$.

The construction you describe when $\mathscr{C}$ consists of all closed sets of $X$ is known as the Wallman compactification of $X$. I'll denote if $\omega(X)$. It is due to Wallman; Lattices and topological spaces, Ann. Math. 39 (1938) 112-126.

Of course some sort of techincal assumption is required.

Let $X$ be a $T_1$ space. Then $\omega(X)$ is a compact $T_1$ space containing $X$ as a dense subspace. Moreover it has the property that every continuous map $X\rightarrow K$ into a compact Hausdorff space $K$ extends over $\omega(X)$. The space $\omega(X)$ is Hausdorff if and only if $X$ is normal, and in this case $\omega(X)\cong\beta(X)$.

Shanin later generalised Wallman's construction; On special extensions of topological spaces, Dokl. SSSR 38 (1943) 6-9, On separation in topological spaces, Dokl. SSSR 38 (1943) 110-113, On the theory of bicompact extensions of topological spaces, Dokl. SSSR 38 (1943) 154-156. The compactifications that Shanin constructed allowed for the ultrafilters to come from more general lattices $\mathscr{L}$ of closed subsets of $X$. Of course at the expense of added assumptions: $\mathscr{L}$ is required to be a so-called $T_1$-base for the closed subsets of $X$. Denote by $\omega(X;\mathscr{L})$ the Wallman-Shanin compactification built using the $T_1$-base $\mathscr{L}$.

Here are some examples to convince you that these compactifications are interesting.

1. $X$ is locally compact $T_2$ and $\mathscr{L}$ consists of all $(i)$ compact subsets of $X$, and $(ii)$ all closed subsets $A\subseteq X$ for which there is a compact $K\subseteq X$ with $A\cup K=X$. Then $\omega(X;\mathscr{L})$ is the Alexandroff compactification of $X$.

2. $X$ is Tychonoff and $\mathscr{L}=\mathscr{Z}(X)$ is the collection of zero sets. Then $\omega(X;\mathscr{L})\cong\beta(X)$, as you have recognised.

3. $X$ is rim-compact $T_2$ and $\mathscr{L}$ is the set of all finite intersections of regularly closed sets with compact boundaries. Then $\omega(X;\mathscr{L})=\mathfrak{f}(X)$ is the Freudenthal compactification of $X$.