Wrong proof about maximum matching
Consider the path graph of length 3: $x-y-z-w$.

Take $S$ to be $y-z$, and $S'$ to be $x-y,z-w$.



A set of constraints for which your kind of argument does work is known as a matroid. The greedy algorithm that you describe does work for matroids (and, more generally, for greedoids). Your set of constraints satisfies a weaker condition: it is 2-extendible, meaning that in order to add a new element (in this case, an edge), you need to throw out at most two existing elements. The greedy algorithm always gives a 2-approximation for such systems of constraints.