I would agree that it isn't a singular issue. Breaking it down between where on the ice the giveaway is happening.
In the offensive zone we see a lot of aggressive pinches and players holding the puck a long time when the "right" play would be to cycle it back into a corner so forwards could get back to work rather than either holding the puck while stopped high in the offensive zone while all the forwards are deeper. Now I say "right" because I view what they are doing to be the wrong play yet so many of them (both forwards and defense) seem content with easing into the soft pocket just inside the blueline rather than driving themselves or the puck deeper into the zone that I view this as the coaching staff wanting all of them to try to make some sort of play rather than putting the puck into space for a 50-50 battle lower in the offensive zone or even toward the net. The resulting turnovers lead to a lot of odd-man rush chances against because they have too wide a gap with their partner (who is often standing still too) and no one covering. So I think system-wise, they play pond hockey and it doesn't benefit them in terms of yielding high-danger rush chances. I would put a lot of these on how they are being coached and there are also times when defensemen activate, pinch, and there is no forward even shading remotely to cover the position. Again, could be "just" coaching, though there are some players who just seem to do it naturally like Benson and Greenway so I think it could be personnel. The kids are my biggest whipping boys on this, but so too is the usual Thompson line where sometimes Tuch will cover but when Skinner does it's usually something that makes me say "holy shit, he did the right thing".
And then in the defensive zone there are a couple of factors - first, the forwards blow the zone early and constantly. That stretches out the space between the defensemen and the forward, making it harder to complete a pass for some (Dahlin and Power show a lot of ability to get things stretched up the ice but there are times when Clifton and Johnson have done noticeably well at it too... the rest seems to be a crap shoot and the four mentioned are also likely to turn things over at times), and also leaves them short handed if there is a turnover. I don't know if this is all systemic or not because guys taking an easier (ie. lazier) path back and not holding the zone until control is assured is the sort of lack of professionalism and effort that bothers me in some of these guys. It requires a little bit more energy coming back and a bit more energy to turn up the ice and transition quickly. So there are fewer options for outlets some of the time.
And then there is the nonchalance factor, the guys making a half-assed pass attempt, or having a lack of urgency to move the puck surely and swiftly. Dahlin's had a lot of this so far this year but all of 'em have their moments too. Power had a lot of these too earlier, seems to have worked some of that out of his game. Unforced turnovers just going D to D that then put them under pressure drives me nuts and that is I feel is personnel.
Then there is the amount of puck watching that goes on, how often people drift slowly out of position, their collective brain farts at how to defend on the rush, and the relative lack of physical engagement around their own net... all things that speak to coaching for me. Perhaps too they are player issues, but I don't look at any of these guys and see improvement in terms of awareness (hey, check off shoulder sometimes to see who is creeping in back door there Joki) and the engagement/battle factor is relatively low on all of their internal guys. The folks they just brought in, yes, there is engagement and sometimes it's bad since they're involved in a battle and pucks go in off of them. But holy crap, watching folks stand around while opponents mine for ore under their goalie's glove or pads makes me honestly yell at the the TV like I'm channeling R. L. Ermey in Full Metal Jacket.
This makes a lot of sense and puts together a lot of details to tell a fuller, more complete story than I had in my head. Your paragraph about the offensive zone issues should really be pinned since it's basically what happens every time you watch a Sabres game, and I suspected that the coach was asking them to do that too.
Remember when we all complained about how Kruger did the opposite? That didn't work either, which makes me think it's the personnel. They give away the puck when they try to make a play because they can't provide adequate support, and they give the puck away when they dump it in because they can't win the battle.
(You really should have a podcast or youtube show or something...you're better than most of them and the main reason I come here.)
I think you're right about the blame distribution here. I really don't think a new coach changes a whole lot, except a new one might not tolerate the lazy play as much (I'm not sure Granato actually "tolerates" it, but that's the popular narrative).
Offensive zone blame: Mostly forwards (can't win dump in battles, can't make a play under pressure, can't maintain possession, can't cover for pinching dmen), some coaching, some roster construction (we're allowed tough guys)
Defensive zone blame: Mostly both forwards (blowing the zone, slow to come back) and defensemen (bad/lazy passes). Significant roster issues (need more Dmen who are willing to punish guys whacking at pucks in front of the net). Some coaching.
Special teams: Granato has always been very slow to make adjustments when things aren't working, but I've heard reasonable reasons for why he doesn't like to make changes often. Not sure I disagree with his logic, but the results are unacceptable. I'll blame him, but the forward, for instance, sure look like they're trying to blow the zone while on a damn PK sometimes, which is confusing.
Goaltending: players are all fine, the only problem is the GM and his inability to asses, and inability to make a decision until it's way too late. I don't blame the coaches either.
Compete: Players. Maybe you can blame the coach for not yelling enough but that stops working pretty quick anyway, if it ever works to begin with.
It's mostly on the forwards imo.