Min weighted edge cover - is the greedy algorithm sub-optimal?
Here is a counterexample to your proposed algorithm:

The optimal solution consists of 3 edges and costs 3+1+1=5, but the min-cost maximum-cardinality matching, which consists of 2 edges and costs 3+3=6, will be chosen by the first step of your algorithm and immediately returned, as it is already a solution. (The only other maximum-cardinality matching costs 1+6=7.)