Back in the day, I climbed that route a number of times. And I descended it several other times after climbing other, harder routes. The area where he fell is steep snow. It is not even remotely 5th class. It is in that gray zone of angle, not really so steep that one feels like they need a belay, but steep enough that to stop a fall you must self arrest instantly. Many climbers solo it, many others do it as a rope team. I don't ever recall seeing anyone belay it, though I'm sure some have. The article linked below estimates that 10,000 people a year attempt Mt Hood, the majority of whom are on this "South Side Route".

It appears that he didn't have an ice axe, but rather was using ski poles with a self arrest pick on one pole. That would not be my choice of tools for the route. With an ice axe, one can push the shaft in at each step, to provide a hand hold to prevent a fall from happening in the first place. And if one does fall, a quick self arrest might save the day. I haven't used a ski pole with arrest tool, but my sense is it is a poor second choice to a good axe.

This article has more details, including time lines.