Mean Square Continuity of Random Process
The whole point of the concept of mean square continuity is to avoid strict assumptions about the continuity of every realization of $X(t)$.

The given proof simply shows that continuity of the autocorrelation function implies that the limit $\lim_{t\rightarrow t_0}E\{[X(t)-X(t_0)]^2\}$ exists and equals zero.

Note that continuity of the mean of $X(t)$ is a consequence of mean square continuity. Since

$$E\{[X(t)-X(t_0)]^2\}\ge E^2\{X(t)-X(t_0)\}\tag{1}$$

it follows from

$$\lim_{t\rightarrow t_0}E\{[X(t)-X(t_0)]^2\}=0$$

that

$$\lim_{t\rightarrow t_0}E\{X(t)-X(t_0)\}=0$$

must hold, i.e., the mean of $X(t)$ is continuous.