Nice addition Mario, as always! Actually, I wanted to remove some graph-related examples and to add other examples. I think talking about maximal matching is going too far :)

But you are definitely right! And the other strategy seems good. I just don't know whether it yields a constant-factor approximation. I'd like to think about it and prove it/disprove it.

The other heuristic does not yield a constant-factor approximation. I do not have the proof, but I've read that it can obtain covers that are log n times worse than the optimal.