I realize this is the pot calling the kettle black as I sometimes also make assertions without attribution in my own writings. However, in this case the entire crux of your argument rests on the accuracy of these two statements. It is fine to say the details are “complex” but to say these results “have been established” is a very, very strong statement of scientific fact, that deserves citation/attribution. For myself there exists a major flaw in the conclusions that I am not sure it is possible to address. In order to run a non-biased experiment assessing the influence or not of belief in free will on behavior one would need experimenters that take no position on the matter (i.e. do not believe or disbelieve in free will). This is simply not possible in my view and is a fatal flaw in almost all ‘scientific’ studies of ‘belief’. Having non-believers design and run the experiments is not good enough, they need to be neither believers nor non-believers in whatever phenomenon ‘belief in’ is being evaluated. This is hard or impossible to achieve in most cases and even harder/impossible to ‘prove’ that it was achieved. Even if a computer is used in the assignment of the tasks the person that programmed said computer would need to meet the same requirements.
I do not question that belief influences/impacts behavior as it obviously does. However, measuring that impact in a scientifically valid way is almost always impossible. This is why it is mostly ill considered in my view to try and use scientific methods to study psychological phenomenon like ‘belief’. That does not mean I believe psychology is totally out of the purview of experimental science, only that some psychological phenomenon are too ‘complex’ to study using these methods. At the very least they are too complex to allow the scientific method to be used to draw strong conclusions about.