If you watch Puljujarvi play against simple opposition, like he did in that little tournament before pre season, it's obvious that he's a playmaker with great vision, and a cerebral player too. That combined with his speed is why he was drafted so high.
In the NHL, fast skating is an asset, but it can also be your downfall because it can just as easily put you out of position, and the result can be that the player becomes timid and just doesn't skate like they could. Also, for a playmaker it can be tough to adjust in the NHL were the gaps close quicker and the checking is tighter. In the AHL, there might be more room but playmakers depend on their teammates a lot so it's a double edged sword. Easier opposition to play against, sure, but not as much talent to work with.
I definitely think he starts in the NHL next season, like he did last season. He'll also have a full offseason to train and work on things, something he didn't have last year because he was recovering from his knee injury. If he doesn't do well again, then no problem. He'll just go to the AHL again in that case and we'll do it all over again the following season. He has the talent and the smarts to meet his potential. It's just a matter of adjustment.