Why does my manual derivative of Layer Normalization imply no gradient flow?
If you're computing the deriviative of the layer norm for the purpose of using it in backprop, then you need to compute the derivative with respect to the parameters of the layer, not it's inputs.
i.e., fix $x$ and compute $\frac{\partial y }{\partial \mu }$ and $\frac{\partial y }{\partial \sigma }$
Regarding your derivation specifically
I did not look at it closely, partly because I suspect that its not actually the derivative you care about as I said above.  But it looks like you're over-complicating things in the sense that you should treat $\mu$ and $\sigma$ as fixed variables and not as functions of $x$.
If they're functions of $x$, it may be that you're running into something like this:
You define $y = \frac {x - f(x)}{g(x)}$ and you compute $\frac{\partial y }{\partial x }$
... quotient rule ... computation... re-arrange...whatever
then you plug $f(x) = x$ in to the result and correctly observe that $\frac{\partial y }{\partial x } = 0$. Which is kind of obvious if you plug in at the start, but has been obscured because of notation. Now, where you started is not as simple as what I've written, but this same thing could happen.