One could argue that this study is showcasing a strategy for people of a certain mindset to improve their willpower; what it doesn't offer is proof that the many other studies that have supported depletion theory use "very shaky" science.
You mean the "certain mindset" specifically selected for by the people running the experiment?
There's nothing in there about the results being limited to "people of a certain mindset".
the results highlight the critical role of beliefs about willpower in self-control performance
I don't know how generalizable the results are, and I'm not making claims about its generalizability. I'm simply refuting the notion that it's wholly incompatible with depletion theory. Speaking personally, I would have appreciated some notion of how common any belief whatsoever is for people.