I think the expression "stopping cold turkey" is misleading.What does it mean?Not acting out?But what if you fantasize each time you see a trigger, yet still manage not to act out? Are you really sober? I don't know... So where is the line that defines sobriety? I think that in the case of lust addiction, it's a moving target. The only thing we can do is works towards "progressive" victory over lust.We may never achieve total victory - but so what?That is not our problem.We can only work, with Hashem's help, on the present.Whether that's considered "cold turkey" is academic.
Weakness and even some measure of failure were - and still are - a necessary aspect in my own recovery so far. If lusting my brains out negates my success at escaping masturbation, then I may as well masturbate and get it over with....
But I understand that you are addressing the fact that what we are really after in SA groups and here on GYE, is freedom from the tyranny of lust, not just technical sobriety. Whether a guy is here to stop doing specific aveiros or to stop ruining his life, we all recognize that just not doing the "problem behaviors" is not enough. We need to get at least some measure of comfort living without our habit. If we routinely lust our brains out - even if we don't "act-out", we are just not there yet. But that's OK! It remains a process.
When it comes to mussar, there are some who vehemently disagreed with Reb Yisroel Salanter's Derech of making tikkun hamiddos the focus and hub of our entire avodah. The Briskers and the Chazon Ish (it seems to me), and most chassidishes believed that our entire tikkun occurs through doing the right thing. Putting our focus on mussar seems to cheapen the Torah. They saw that through being moser nefesh for keeping halocho with dikduk, ahavah and yir'ah - no matter how difficult, we will automatically become a Jew with Yir'ah, bitul to Hashem's Will, bitachon in Him, and respect and love for his fellows. They'd say (at least as far as I understand them) that our tikkun grows out of the avodah itself. And our tikkun is best left in Hashem's hands, lest the entire Torah and avodah becomes a self-centered - albeit noble - pursuit.
For what it's worth, the 12-Step Program works exactly this way in my life as well, i.e. not in the mussar way. For example, the main priority for me is not to act out. To act out would open the floodgates and ruin my life at this point. That is not up for grabs. But it turns out that I was forced to work the steps only in order not to act out! I'd have resentments and fears that were eating me up, or just feel uneasy... I'd sit on it and go crazy - till it became clear that if I don't do something about it, I will act out. So out came the Big Book, the chart, the calls, and there is no choice but to work steps 4-7 on it. And life gets better.
Is lust is becoming a problem for me? - I'd sit on it like an idiot till my gauge was in the red zone and finally out would come the 3rd step prayer, the calls, and the Big Book - I'd search for the 'secret' of the 3rd step, cuz my heart tells me that it's the answer to everything. I'd then give up whatever I could of me to Hashem again - all so that I don't act out and lose my life. I'd focus on being attentive to my wife and kids to really learn how to care about them - so that I do not stay a selfish, self-centered man, because selfish bums are guaranteed to act out, you know...
Now, you'd think that if a guy really believes that his main priority is to not act out, he'd be focused on not acting out... I hope you can see that is not at all the case. "Not acting out" is no longer my main preoccupation - being a decent man, staying free of lies, and focusing on being helpful to G-d and man - that's my preoccupation.... (on a good day!)
Well, I went on for too long, as usual. Sorry. Uh-oh, by now our turkey is really cold! :-)