How to represent a 0-valid boolean formula?
The constraints of a $0$-valid formula can be more general. All we know about them is that they are satisfied if all variables are $0$. For example, a constraint could be $x = y$ (i.e. $(x \land y) \lor (\lnot x \land \lnot y)$). Your form is not general enough, since it only considers constraints of the form $\lnot x_1 \lor \dots \lor \lnot x_\ell$.

While your formulation doesn't capture the problem of determining the most positive solution for a $0$-valid CSP, it is equivalent to an even more classical problem, set cover. In particular, your problem is NP-complete.