I don't think coaching is the only issue here, in fact I think its the smaller issue with respect to what is really ailing this team.
Not a fan of Eakins in any form, but lets think about it. Over an 82 game schedule, I think its reasonable to believe that the coach will try new things, especially when losing. This is after the fact that a system or method has not worked.
I think it would also be assumed that these guys have video of other teams, especially those with "good" defensive systems. So over the course of a season, its not really a secret what the defense (and forwards to a lesser extent with respect to this argument) are doing on successful teams in basically every situation that appears in a hockey game.
So in this sense Eakins is really responsible for employing a system that best matches the talents of the team, motivation, and accountability of the players. Motivation is the hardest to tell from our perspective so I'll leave it out.
Accountability -- which we can witness from scratches/benching/minute reducing. In my opinion we do not see this enough, it seems like some players have a much shorter end of the stick and see their minutes reduced when they screw up versus when other players do. I think this is flawed because I think if you **** up, you **** up. Whether you're a star/ veteran/ rookie it should be the same. If players see that others are going to be treated differently its going to create a division, they already know they get paid different, now they get treated differently, it just doesn't send a good message.
Now as for benching in a winnable/close game with not much time remaining I can definitely see why you would treat your stars differently, but over the course of the season this kind of behavior should be exercised in "crucial games". I think players would see the fact that certain players need to be out regardless of mistakes late in the season versus the start of the season.
Now as for the system, whether the system is good or bad, the accountability of players will play a factor here. Follow it, rewarded, do not follow it, punished. Simple as that. I think this plays a big part in why we see our players all over the place, I doubt Eakins instructs them to be all over the place, out of position, or not taking assignments.
So if this is true, this mean that players are A. not being held accountable enough, or B. not talented enough to do there jobs given. Looking at the defense Eakins has to work with, I think most would agree that this team does not have a 1 or 2 defenseman on its roster, it has a bunch of 3,4,5,6 defenseman. So even if the team has accountability and great system, there are bound to be more mistakes simply by virtue of lack of talent on the defense.
What is really ailing this team is the lack of top end talent on the defense. Why? Because teams with terrible systems can over the course of a season change their systems accordingly to mimic successful ones and can at their discretion improve accountability. This means that no matter what happens, on average a team will always be at a disadvantage when it lacks top end talent on defense.
I think of it like, you can give a person all the wood, a fire pit, even some paper but unless there is a spark you're never going to see a fire.