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 https://www.cl.ca

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 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.