By La Salle's Invariance Principle, $\omega(x) \subseteq M_c$ for some $c$. The set $S = \{y \in D : \dot{\mathcal{V}}(y) = 0\}$ is just $\{0\}$ when $\mathcal{V}$ is strict. Hence $\omega(x) = \{0\}$, meaning there exists a sequence $t_n \to \infty$ with $\phi_{t_n}(x) \to 0$. By Lyapunov's First Theorem, for each $\epsilon > 0$ there exists $\delta > 0$ such that $|z| < \delta$ implies $|\phi_t(z)| < \epsilon$ for all $t \geq 0$. Since $\phi_{t_n}(x) \to 0$, there exists $N$ such that $|\phi_{t_N}(x)| < \delta(\epsilon)$. Applying the Lyapunov stability estimate with $z = \phi_{t_N}(x)$ gives $|\phi_{t + t_N}(x)| < \epsilon$ for all $t \geq 0$. This holds for every $\epsilon > 0$, so $|\phi_t(x)| \to 0$ as $t \to \infty$.