Branch cuts are an artefact of people who insist on wanting to graph a function on the complex plane when really they shouldn't.
Branch cuts are ugly.
A branch cut is not an essential property of a complex function.
How can you be sure that when you talk about "$\log(z)$", your interlocutor is using the same branch cut as you ? You can't.
The radius of convergence is a local property of a function, it is the upper bound on the set of radii for which you can analytically continue the germ of your function on an open ball centered around your point. But if you make bad enough branch cuts, the poles, singularities, and branch points that you see may not be the ones relevant.
It turns out that when I have to choose a branch cut for $\log(z)$, noone is stopping me from choosing the curve $\gamma(t) = te^{it}$ for $t\ge 0$.
When I plot the function $1/\log(z)$ (with the branch of $\log$ where $\log(1)=0$), and look at the point $z=7$, well I have a branch point at $0$ and
a pole at $1$, so the radius of convergence should be $6$ right ? Wrong ! the radius is $7$.
In fact you can do even worse, for example if I take the branch where $\log(1) = -2i\pi$ and plot the branch of $1/\log(z)$ with my branch cut, the radius of convergence at $z=7$ looks like it should be $7$ because I only see a branch point at $0$ and no pole, but in this case the radius is actually $6$ !
So really, stop thinking in terms of branch cuts.
You should think of the graph of "$\log(z)$" as the subset $G = \{(\exp w,w) ; w \in \Bbb C \} \subset \Bbb C^2$. This subset looks like a graph of a holomorphic function if you don't look around too much, but when you look at it globally, it turns out it isn't.
But, this subset still allows you to talk about the radius of convergence of the function it looks like locally at a point $(z,y)$, as the least upper bound of the set of radii $R$ for which the connected component of $(z,y)$ in $G \cap B(z,R) \times \Bbb C$ is the graph of a holomorphic map $B(z,R) \to \Bbb C$.
And even then, sometimes the continuation of the graph is not a nice enough object to study the analytic continuation of a function, that's why we eventually need to use general complex manifolds $M$ with canonical maps $z : M \to \Bbb C$. Then a function on $M$ is a holomorphic map $f : M \to \Bbb C$ and you can "plot" $f$ by plotting $(z(m),f(m))$. In the case of $\log(z)$, $M$ is just $\Bbb C$, $f$ is the identity, and $z$ is the exponential map.