The justification for why that formula is correct (equivalent to the fundamental definition) is even more incomprehensible than the formula itself if written in English, but fairly clear, almost trivial, using algebraic notation. That is, if you want to "explain both the why of this function and how this specific function carries out that logic," using algebraic notation is far, far better than just using a natural language.
√(Σ(xᵢ²/N) - (Σxᵢ/N)²) is unhelpful on its own, and as you point out a more "human readable" version of this equation would be rather convoluted. The middle ground here would be to use more specific syntax like yours, while also ensuring that each variable is clearly defined and this logic is robustly documented to explain both the why of this function and how this specific function carries out that logic.