Improving set intersection lexicographically

For me convergence of the proposed algorithm sounds too optimistic and indeed it can get stuck even in the following special case.

Let $G$ be a graph with the set $V$ of vertices. Let $A_1,\dots, A_n$ be edges of $G$ considered as set of size two. If $G$ has a vertex cover of size $k$ then the required minimum is at least one. On the other hand, the algorithm can get stuck in this case. Indeed, let $V$ be a disjoint union of set $V_1$ and $V_2$ of size four each. Let every vertex of $V_1$ is adjacent to every vertex of $V_2$ and the subgraph of $G$ induced on $V_1$ is a cycle of length four. Then $V_1$ is a vertex-cover of $G$. On the other hand, the algorithm get stuck at a set $V_2$, because it covers all edges of $G$ but four edges of the cycle, whereas each four-element subset $B$ of $V$ with $|B\cap V_2|=3$ covers all but five edges of $G$.