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

The reason is that $r(t)$ and $\phi(t)$ are your independent variables, so you are not allowed to use the chain rule for them, because $\phi$ does not depend on $r$. An analogy would be the following: \begin{equation} \frac{d}{dx}x=1 \end{equation} However, if I introduce another independent variable $y$, it would be of course wrong to write \begin{equation} \frac{d}{dx}x=\frac{dy}{dx}\frac{d}{dy}x=0 \end{equation}

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

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

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.

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.