I've been doing a lot of reading lately about Shiva Nata, because I've recently started learning it and I am curious about what other people have discovered. Your question and some of the comments here reflect some of the same concerns I had before starting, but I managed to reason my way into convincing myself to give it a try.
Yes, the Shiva Nata site portrays the practice in an extremely woo-woo way, but reading past that (and reading other posts on that site and others) I managed to extract some useful tidbits that were genuinely fascinating to me. I was drawn in to the idea of having a new physical practice that would challenge my mind, especially since the practice involves number patterns and I am highly mathematical by nature (I am a grad student in theoretical physics). I was extremely intrigued about the claim that the practicing of the patterns of movement could help me get a better feel for connections between ideas (which seems to be what the author of the site calls 'epiphanies'). By 'connections', I just mean the most general sense of the word: any association, or similarity, or opposition, or orthogonality, or way of describing ideas relative to each other.
The practice itself seems to be fairly unique in how its purpose is to trip up the mind: you learn a pattern, then you learn something that mixes up what you had learned originally. When your mind reassembles the lessons into a new way of remembering both, both lessons are strengthened. By extension, the mind's natural ability for pattern-matching is strengthened as well, and this ability can be applied outside the practice as well.
Ideas can be examined in many different ways. Since most ideas have multiple parts or perspectives, it seems reasonable to consider that it could be beneficial to look at the parts of the ideas in different combinations and different orders. It's one thing to just try and see things in a different way, and completely another thing to have a physical sensation of how many different ways things can be done or seen or experienced or analyzed or combined. If the mind has a natural tendency to notice patterns (and this is a claim that has been substantiated by research), and if the body is used as a way to explore variations in patterns, it seems to make sense that physical movements could be used to teach the mind more ways to look for patterns. And how could this not have some sort of benefit to analytical capabilities?
It also seemed reasonable to me to consider that this practice might be able to help me better analyze things about myself. I had already considered myself to be quite skilled in this respect, but there are still behaviours that repeat despite my attempts to understand them and talk myself out of them. However, if the Shiva Nata practice is geared towards seeking new ways of looking at patterns or changing how to interact with them, it seems very possible that it could help me develop insights and techniques to understand and change the patterns in my behaviour, or the ideas that are behind the behaviour.
I'm sure any mental exercise can help with self-analysis and problem solving skills, and any physical exercise can be beneficial to the mind as well as the body, but this particular practice seems to focus on the mental benefits while the physical benefits are just a convenient side-effect. Maybe there are other activities that do this too, but I am quite pleased so far with my explorations into this one. I am still just starting out, but I already feel a lot less stuck in my behavioural patterns, and I think my memory is even getting a good workout, not to mention my arms. Even if some of this is due to the placebo-like effect of "thinking that what I'm doing is going to help", it is pretty convincing, and I have no problem just appreciating the benefits and continuing to apply the practice that seems to encourage the benefits to continue.