Very true and, if the USMC is correct, disappointing.
OTOH, there were several involved that knew for sure whether Bradley or Schultz was involved and they didn't say anything either.
It seems possible that at one point Bradley in fact had his hands on the flag/flagpole. It also may be Schultz was a recent replacement who didn't survive very long and that Bradley was highly respected. IIRC, they were taking sporadic fire and the flag raising would not have been uppermost on their minds; that some were posted as a defensive measure and that would not have included Bradley, i.e., it would have been natural for him to serve as a flag raiser.
Schultz is a german name. Maybe they didn’t want that getting out.