I think it's an interesting area! But the studies referenced in the article are 50-70 years old, and I wasn't able to find them on google scholar or any of my university's journal databases. Without being able to read the studies and evaluate their design (which I'm assuming is probably not the best given how old they are), I'm with a few others who have commented in not really believing it. From a cognitive psychology perspective, the language used in the linked website is pretty layman in it's explanations and terminologies. Auditory neural pathways and how they intertwine with language and motor functioning is not that simple, but it would be really cool to reopen this area of research if it hasn't been already. There definintely is a connection though between auditory feedback and stuttering. I once read about a trick where you stop stuttering if you listen to music to drown out your own voice. I've tried this multiple times with class presentations and it definitely worked for me. Total fluency!