Joe Rosenthal's Iwo Jima flag-raising misidentification - How?

Story today. In short, in the second, better known, flag-raising photo, Pfc. Harold Schultz participated, and not Corpsman John Bradley. Bradley had indeed participated in the first raising.

The GQ: It seems as though Schultz and Bradley would have known which raising they had participated in. Did they ever speak on it? They both lived about another 40 years, so they surely had opportunities, so it’s somewhat odd that it is only confirmed 20 years after their deaths.

?!?? They couldn’t be bothered to link to the less-known image as well, for comparison?

Here’s a page discussing the evidence and showing the photos.