I think the NYR actually dump the puck plenty. They get to it often. But then they have no clue what to do next except rotate the puck and themselves around a 3ft perimeter of the offensive zone along the boards. Yesterday their best extended period of play was a 2-shift stretch (lol) in the second where they were cycling well and the kicker, really using the dmen well (Lindgren and Schneider were on I believe) to move the puck creatively among all 5 players. Then Zac Jones came on, did a really nice job in the neutral zone to spring Zibanejad back into the zone with a head of steam.
To me what they don't do well, that many of the good teams do well (Canes, Panthers, Avs for sure do this) is carry the puck with speed, under great control, into the offensive zone, as an (at minimum) 3 man forward unit. Watching the Canes, their forwards just constantly control the puck from the red line, over the blue line, as a 3 man unit and then create a 3-on-2, 2-on-1 with a net crasher type of situation where the goal is simply to get towards the middle of the ice in space and get a shot. This also requires team cohesion and execution from the dmen to spring their forwards with speed.
The NYR too often get bad breakout "passes" i.e. borderline blind clears into space, forwards are disjointed and not moving in unison, too many forwards are just flat out too slow to execute this (Kakko), too bad of puck handlers to execute this (Kreider, Chytil, frequently Lafreniere, Hunt), or intentionally choose not to play this way and intentional slow the game down near the blueline (Panarin, Strome). Zibanejad is one of the few players the NYR have who can play like this consistently. It's an issue that transcends roster construction, team culture/dynamics, and pure gameplan x's and o's.