What can go wrong with applying chain rule to angular velocity of circular motion?

$dphi/dr$ is undefined. $phi$ is only defined for 1 value of $r$ and is thus not differentiable. So you end up with $omega$ being zero x undefined quantity.

Though $frac{mathrm{d}r}{mathrm{d}t}$ is $0$, $frac{mathrm{d}phi}{mathrm{d}r}$ also tends to infinity. So this is a zero into infinity form and its limit will result in a finite quantity which will be equal to the angular velocity calculated without chain rule .

$dr/ dt ≠0$ as direction of $r$ is continuously changing. $r$ means both direction as well as magnitude, you are only considering the magnitude, which is wrong.

The meaning of pushing an intermediate variable is to relate the dependencies of change. So, the expression

$$ frac{dphi}{dt} = frac{d phi}{dr} frac{dr}{dt}$$

Assumes that the angle $(phi)$ can be written as some function of the radius from the origin i.e: $phi(r(t))$ but is that really possible in this case? We can say that there is no 'differentiable' map from $r to phi$.

I mean think about it, how would you construct a function which associates the radius , a fixed variable, with the angle which changes with time? It would not even be a function because you would have to span all the angles with a single radius value.

The problem is that $r$ is a constant, so its rate of change relative to any other variable must be zero. Thus $$frac{text{d}r}{text{d}t} = 0$$ and $$frac{text{d}r}{text{d}phi} = 0$$

So $frac{text{d}phi}{text{d}r}$ is undefined. When you try to apply the chain rule in

$$omega = frac{text{d}phi}{text{d}t}=frac{text{d}r}{text{d}t}frac{text{d}phi}{text{d}r} = frac{text{d}r}{text{d}t} / frac{text{d}r}{text{d}phi}$$

you're attempting to divide zero by zero, which is an indeterminate form.

Since you've got vectors on your diagram, I think it's worth noting that you could write $$omega = frac{mathrm{d}phi}{mathrm{d}t} = frac{mathrm{d}vec{r}}{mathrm{d}t} cdot vec{nabla}phi$$ because $vec{r}$ is a function of $t$.