= Solution
Suppose that a connected smooth surface $\widetilde S\subset\mathbb R^3$ properly contains $S$. Since $S$ and $\widetilde S$ have the same dimension, $S$ is open in $\widetilde S$. Choose a point of its relative boundary. Inside a sufficiently small compact <normal neighbourhood> in $\widetilde S$, choose $q\in S$ and a closest point $p\in\widetilde S\setminus S$. A minimizing $\widetilde S$-geodesic from $q$ to $p$ stays in $S$ before its endpoint, since encountering the complement earlier would contradict the choice of $p$.
The restriction of this curve is consequently a geodesic of $S$ with a finite maximal endpoint at $p$. On the other hand, the <Gaussian curvature> of $S$ is the restriction of the smooth Gaussian curvature of $\widetilde S$, so it remains bounded near $p$. This contradicts condition $(*)$. Hence
$$
\boxed{\widetilde S=S},
$$
so $S$ is an <inextendible embedded surface>. This is the <curvature-blowup criterion for inextendibility of a surface>.
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