Of course, this is not correct. As a simplest example, let $X$ be a random variable which takes only values $\{0,\ldots, n\}$, then the moment generating function is a polynomial of $e^t$, therefore its radius of convergence is infinite. But $\log M(t)$ has finite radius of convergence since a polynomial of degree at least $1$ has some zeros in the complex plane.

In general, the radius of convergence for $K(t)$ is the distance from the origin to the closest zero of $M(t)$.

Of course, this is not correct. As a simplest example, let $X$ be a random variable which takes only values $\{0,\ldots, n\}$, then the moment generating function is a polynomial of $e^t$, of degree $n$, therefore its radius of convergence is infinite. Any polinomial with positive coefficients which add to $1$ can occur. But $\log M(t)$ has finite radius of convergence since a polynomial $P$ of degree $n\geq 1$ has some zeros in the complex plane. So $P(e^t)$ also has zeros, unless $P$ is a monomial.

In general, the radius of convergence for $K(t)$ is the distance from the origin to the closest zero of $M(t)$.