Jack Z is by no means perfect, and has made some head-scratchers in his time, but 'can't-miss' prospects busting is not really a GM's fault. Your job is to acquire as much talent as possible in a way that puts your team in a position to succeed. What makes one prospect pan out and another fail? It's arguably unlikely to have much to do with the GM (I'm sure there are arguments to be made to the contrary, but bear with me).
Players like Ackley, Montero, and Smoak were considered LEAGUE-WIDE to be the best in baseball. People were AWED that we were able to somehow get both Montero and Smoak into our farm system, and Z was worthy of the accolades. They've struggled, and sure, that obviously is going to reflect negatively on Z, but I would argue that it shouldn't. If Bedard had gone on to win 5 Cy Youngs with the Mariners and Adam Jones et al had flamed out, it still would have been a bad trade. Sometimes bad moves work out, and sometimes good moves don't, but hopefully you hire someone who consistently makes good decisions that give you better odds of success.
I think of it like poker - you can go all in when you draw a 2/7 off-suit and win, but it doesn't make you a genius. It means you made a bad decision and got lucky. You can go all-in with pocket aces and lose. It doesn't mean you're dumb. It means you made a good decision and got unlikely. (I know there's a lot more to poker than this, but the metaphor holds).
Again, Z isn't perfect. He's done some things I really disagree with. But whereas Bavasi was making bad decisions which turned out bad, Z is making (generally) good decisions which aren't working out. The latter process is more likely to eventually yield results than the former.
I'm not saying Z should definitely keep his job, but I don't think he should definitely be fired. I would say the same for Wedge (though I don't like his managing style for a variety of reasons, I think he has little to do with the team's struggles). Discussions about ownership/Lincoln/Armstrong have been had and I won't rehash those.
As for the coaching staff and how they impact the development of prospects, it's really hard to say. The Mariners have sacrificed SO MANY hitting coaches and managers as scapegoats in this decade of suck that I find it hard to believe they have much to do with it. But it's certainly not out of the realm of possibility.