It is impossible to know. I don't think most coaches go into it for egotistical reasons. At the initial levels, it's much like teaching, I think.
But yeah, ego can intervene once success is attained and maybe is something of a prerequisite for success.
But most people who attain sobriety are in a bit of a quandary. Their lives have gotten messy, some of their life choices, including bit things like career have become questionable. Something you hear a lot in AA rooms and in discussion among addicts is this:
Your best thinking so far has gotten you here, maybe it's time to try something different.
So, just about everyone in recovery considers whether their "old job" is compatible with sobriety and what they really ought to be doing. And, sometimes career changes follow. But sometimes they don't. Most everyone probably takes a stab at resuming their old career with some new approaches to it that come with sobriety and sometimes it works out swimmingly, sometimes it doesn't.
This includes careers like lawdog, where sometimes being an asshole dickface is part of the job, or can be. The newly sober lawdog has to consider and find out whether the job can be done well, and with reduced stress, without being an asshole dickface motivated by ego and insecurity.
I think Sarkisian found himself in that spot and Saban's "rehab program" was the ideal place for him to discover whether he could continue to coach on a "new paradigm" so to speak. And it seemed like he could.
It would appear, outwardly, that Sarkisian is succeeding with this new paradigm that is less ego-driven and more akin to the teaching aspects of the job, the "being of service to others" aspect that is so critical to maintaining recovery and sobriety.
ETA: Sark was going to be a better person as a result of recovery. Whether that better person could be a top-flight HFC at a top-flight program remained to be seen. It seems like he's putting it together in a way that his recovery is positively influencing his coaching. That was not a sure thing.