I think Jesses best fit is being the offensive threat on a 4th line.
He has some offensive tools, but mostly rushing down on the wing firing the puck.
I would sign him as a 4th liner at 1 million as a job to forecheck the puck, try get some scoring chances, firing the puck, turn momentum. And has to ptove himself 3rd line worthy if to climbing higher in the lineup.
Might be his "best fit"...but it's still not really a good one. If he could do that effectively, at least take possession of the puck, carry it up ice and fire a pointless puck on net to get an offensive zone faceoff for some other line...that'd be alright.
But without really talented support to shoulder a lot of the load, he doesn't seem all that reliably capable of even getting
that done when on a bit of an island by himself. His shooting is also erratic, which means it's not always just an automatic faceoff crest sniper exercise. It can also be a thousand feet wide which turns around into an opportunity the other way. Really good "grinding" 4th liners, you want to be able to sustain a little bit of a cycle, get to the net and manage the puck in a consistently "safe" controlled way. Including a lot of dump-ins, retrievals, cycle play, over one-man-army stuff. Puljujarvi doesn't seem smart enough, or in control of what he's doing enough, to reliably perform that way. Nor does he seem to have the mentality or vision to assess when to cut bait and just make the risk averse "boring play", or the agility to change course and adapt to that on the fly.
Moreover...he's not
really an "offensive threat". He doesn't seem productive at all there...even to a respectable 4th line level. Which makes him less useful than other "offensive 4th liner" options if you want to go that way. Nor is he bringing the defensive zone start, PK, responsible sort of utility that you'd want from a 4th liner in the other more conventional mold.
He's basically a "tweener". Not nearly skilled enough for the Top-6, but not reliable or responsible enough with the sort of game to thrive in the Bottom-6. You'd think with his tools, he should be able to apply them to become like a bigger, Tyler Motte type. But it's the tools he's missing, and the toolkit he
doesn't have that still hold him back, even in a "grinder" role.