I still call BS on the "Hynes' system is fine" and "the coaches know all the systems and schemes and just have to get their message through".
Hockey is actually a highly-strategical game, although less so in NA where the ice surface is much smaller.
I'll dig into few, most pressing problems we have in my opinion:
In the DZ, our coverage is actually good from the point, as you can see from the graph below. What's the problem here is that our center is usually covering the center of the ice in a wide range, and that leaves the center often vulnerable as the wingers are not quick enough to cover the center going for coverage up the ice. That's why we're allowing so many (quality) shots from the low-slot.
How to fix this problem, then? I'd start with implementing a stricter zone defense policy, which might lead to longer cycles against but limits quality chances for the opposition from the best scoring spots. This is also a more energy-conserving strategy which could lead to more dangerous rush chances for us.
You could also go with a more man-to-man defense, but I'm afraid our slow-footed team would get eaten up against faster teams in that scheme. A more covering, a more stagnant zone defense is the better way as we also need to slow the pace of the game down - something we haven't been able to do under Hynes and that is why the series against Colorado last playoffs looked so ugly.
If you want slow the game down (and we should slow the game down as we are a slow, heavy team) you need to clog up the NZ. Well, how are you going to do that with two of your wingers hounding the puck in the OZ and getting constantly left behind the play? That also leaves us with the problem of the distances between the five-man unit getting way too long. If you look at Carolina for example, you usually see all their five guys within the same frame - an event that rarely occurs with us.