That "really works" part is why approximately 80% of western therapists practice CBT (cognitive behavioral therapy) which was pioneered by Albert Ellis and his team after inspiration from capital S Stoicism. Not psychoanalysis or any of the other popular 20th century approaches to the mental health of modern man, but a derivative of Stoicism that has proved very powerful.
As someone who has been reading Stoic works for 5+ years and for the last two years practicing my own derivation on a daily basis, a part of me agrees that the conflating of the modern adjective stoic and the philosophy and approach of Stoicism is indeed unfortunate. However, I do not believe that this is the primary barrier to entry, but rather the fact that practicing the philosophy is hard, slow work. Just as effective CBT asks a lot of the patient, Stoicism is not a quick fix. In some ways I see the misnomer as a tiny hurdle to entry that weeds out those only interested in dialectic rather than the hard work of changing and crafting one's character.
I would be dishonest if I didn't also point out that Ancient Stoicism has some fundamental problems that are also likely a barrier to the thoughtful. One of its founding pillars is a teleological universe which, thanks to advances in science, we now know to be false. Further, Stoicism glosses over aspects of human behavior (such as addiction and trauma) and is outdated in its approach to inter-community relations, approaching social contracts in the context of small, warlike city states (and later in the context of an authoritarian, imperialistic empire).
A modern, widely adoptable descendant of Ancient Stoicism would be quite different and most certainly wouldn't be called Stoicism. In the end, I don't believe the layperson's misunderstanding of Stoicism needs to be fixed. Truth needs no defense - the predictive and eudaimonic power of Stoicism is there for anyone to pluck if they have the impulse to reach for it and the courage to do the hard work of applying and reforming the self, along with the philosophy.
That said, it's awesome, you all should check it out.