Let's be clear about what we mean. Let $\mathcal{C}$ be a category, not necessarily small or with pullbacks. A coverage on $\mathcal{C}$ is a collection of families $K (C)$ of sinks on $C$ for each object $C$ of our category $\mathcal{C}$ satisfying this condition:
- For every sink $\mathfrak{U}$ in $K (C)$ and every morphism $g : D \to C$ in $\mathcal{C}$, there is some sink $\mathfrak{V}$ in $K (D)$ such that, for each morphism $h$ in $\mathfrak{V}$, the composite $g \circ h$ factors through some morphism in $\mathfrak{U}$. (More succinctly, the sink $\{ g \circ h : h \in \mathfrak{V} \}$ is subordinate to $\mathfrak{U}$.)
A saturated coverage on $\mathcal{C}$ is a coverage $K$ on $\mathcal{C}$ satisfying these additional conditions:
For each object $C$, the singleton sink $\{ \textrm{id}_C : C \to C \}$ is in $K (C)$.
Given any sink $\mathfrak{U}$ in $K (C)$ and any choice of sinks $\mathfrak{V}_f$ in $K (\operatorname{dom} f)$ for each $f$ in $\mathfrak{U}$, the family $\{ f \circ h : f \in \mathfrak{U}, h \in \mathfrak{V}_f \}$ is also in $K (C)$.
Given $\mathfrak{V}$ in $K (C)$ and any sink $\mathfrak{U}$ on $C$, if $\mathfrak{V}$ is subordinate to $\mathfrak{U}$ (i.e. each morphism in $\mathfrak{V}$ factors through some morphism in $\mathfrak{U}$), then $\mathfrak{U}$ is already in $K (C)$.
Given a Grothendieck topology $J$ on $\mathcal{C}$, one can define a saturated coverage $K$ by setting $K (C) = \{ \mathfrak{U} : \mathfrak{U} \text{ is a sink on $C$ and generates a sieve in } J (C) \}$ and conversely, given a saturated coverage $K$, one can define a Grothendieck topology $J$ by setting $J (C) = \{ \mathfrak{U} : \mathfrak{U} \text{ is a sieve on $C$ and is in } K (C) \}$ and these constructions are mutually inverse.
Your question is how to cut Grothendieck topologies out of the picture. This is simple enough: given any Grothendieck pretopology $T$, we can build a saturated coverage $K$ as follows: $K (C) = \{ \mathfrak{U} : \mathfrak{U} \text{ is a sink on $C$ and, for some $\mathfrak{V}$ in $T (C)$, $\mathfrak{V}$ is subordinate to $\mathfrak{U}$ } \}$ If I have not misread your symbolic formula, this is precisely the construction you have suggested. This is because the relation "$\mathfrak{V}$ is subordinate to $\mathfrak{U}$" is equivalent to "(the sieve generated by) $\mathfrak{V}$ is contained in the sieve generated by $\mathfrak{U}$", and it is well-known that the family of covering sieves in a Grothendieck topology is upward-closed.