The pictures seem to bear out this account.
The second is the shot in its original state, before it was tight cropped.
caption notes that these are 5th Div. Marines taking down the first flag while the second flag is secured in the background.
Not everything is a photo-op, sometimes you just have a better photographer present at one time than another, and he gets a better picture.
I didn't think he just happened to be present, maybe just happened to be present when they decided on a bigger flag, but my understanding is it was a bit of a hike to where they put the flag and nobody would just happen to be there.
I wouldn't say not everything is a photo-op, as you point out a good photographer can find the right shot in just about everything. I'd say that not all photo-ops have little pieces of tape on the floor (or a mysterious pile of stones on the beech as it were), and those tend to be the better ones.
Thank you for posting that. Most of it is correct, except that Rosenthal really caught that shot by accident - he swung around and snapped without getting a good look. He never actually attributed the photograph to his skills.