Alright, so normally this is the part of the conversation where I'd be all dismissive and sarcastic and attempt (probably unsuccessfully) to make you feel 6" tall all for some cheap laughs at your expense. Instead, I'm gonna go another route, because I just saved 15% on my car insurance by selling my car.
You're right, that the coach and roster are the same. So, the first thing you have to ask yourself is this:
1) If it's a coach vs issue situation, who is more easily-replaced, 20 players, or 3 coaches? Even if you are willing to save a handful of players, even as many as 10, you're still talking about attempting to unload the dregs of your roster in a league that has become MASSIVELY CONSERVATIVE when it comes to making trades. Now you're asking the rest of the league, who ALL still believe they're still in the playoff race because it's only November, to take your bad players and their bad contracts while giving you something of value (even if that something of value is simply "not cap back"). Meanwhile, a legitimate star coach is sitting unemployed.
2) What is more likely: a team, whose roster has been dominant for stretches THIS SEASON and whose core has won multiple cups suddenly becomes bereft of all speed, skill, intelligence halfway through the season, OR that the team has found its way into a funk (like most teams do) and the organization has tried everything from yelling and screaming to trading a top-6 roster player to get them out of it, to no avail, and at some point you need to wonder: has the coach lost the ability to reach these guys?
You can say roster! roster! roster! all you want, but I don't understand how you can say it and not even once go coach? coach? coach?