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Elliott Wave Theory (EWT) "works" because market behavior is self-similar, or fractal-like, as the result of the "rhythmical" way that traders behave, as @Phreakerx pointed out. It, and traditional Technical Analysis (TA) in general, were developed as heuristics before we had the computing power, as well as the mathematical sophistication, that we have today. It's important to remember that these indicators were developed so that individuals could plot them out by hand, so the procedures had to be simplified to a degree.
Without cheap and efficient computers, there was no practical reason for Elliott to ground his theory more than he did. To some extent you aren't wrong in thinking it's "mostly statistics", because contemporary Quantitative Analysis (QA) uses a lot of the same core ideas as EWT and TA, except now we have the ability to build complex computational models. Because of this radical shift in theory and technology, it's difficult to assess whether EWT or TA is effective in principle, because trying to use those tools in today's environment is kinda' like bringing a knife to a gunfight.
As a footnote, Benoit Mandelbrot, who developed the idea of fractal geometry, was even accused of plagiarizing Elliott, to which he responded, “The idea is ancient, but his use and mine stand in absolute contrast”. You can learn a bit more about the Elliott-Mandelbrot fiasco here: https://blogs.scientificamerican.com/news-blog/mathematicians-predicted-stock-mark-2008-09-16/
