If I'm reading this "shooting score" correctly it seems to mean that when the puck was on Nico's stick on the PP, he shot better than you would expect an average PP shooter to shoot. If that's the case then it seems consistent with what I saw. He wasn't the featured guy, but when he had opportunities he was very good at cashing them in.
Jack's and Timo's numbers make sense here too. They are both volume shooters. Yeah, they both score a lot of goals, but they also both take a lot of shots. If you give a shooter time and space on the PP then eventually the puck will go in. That doesn't mean it was particularly impressive by the shooter. The difference is that Jack is still an elite distributor while Timo doesn't seem to do very well in that area.
Bratt not overperforming the field as a shooter makes sense too. He may have had the highest quality opportunities on the PP given where he say on the far wing. He benefitted from a lot of royal road/pass-off-pad set ups that gave him lots gaping empty nets. So again, he's just feeding off high quality being generated by other players, good goal production but not really special in any way. He's doing what you would expect any PP shooter to do if they consistently had empty nets to shoot at.
Dougie is the one that really confuses me being squarely in the passer category. Obviously he gets a ton of PP assists being the QB that's sitting at the dot. But he was tied with Josi and Ekblad for 1st in the league in PP goals among defensemen and he did it with fewer shots. He also scored more goals on fewer shots than Nico. It looks like he was one of the best blueline shooters in the league last year, yet has a below average shooter score. Maybe that's just a feature of him being a defenseman in a sample that's an aggregate of mostly forwards, so he looks worse by comparison?
But yeah, calling Hischier "the main shooter" is definitely off the mark, but I think the point is that his shooting was the most efficient of anyone on our PP when you factor in quantity, quality, set-up, and finishing.