Your intuition was right in that explanation provided in this proof is incorrect.. https://www.cl.cam.ac.uk/teaching/1415/AdvAlgo/lec8_ann.pdf has a more complete explanation. The algorithm works (and gets a 2x bound), but only because of how it processes the edges sequentially and makes sure to only process edges that aren't already covered.

Assuming your star is a single center vertex which is a vertex to all edges (like spokes in a wheel), then the optimal cover for all edges is 1, the central vertex. If you follow the algorithm, you will select two vertices and produce a covering, which is within a factor of two of the optimal.

I remember proving this algorithm in class (a while ago). What I found interesting from Wikipedia is that this is the best known approximation, there is a proven lower bound of 1.363.

If you pick up a leaf, you also pick up the star one level up and thus the entire subtree is now covered (so you don’t look at it again). What is the failure, specifically?

True. Hmm.