The equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ for a complete metric space $X$ follows from two facts:
for any $\tau$-additive measure $\mu$ on $X$ its support $supp(\mu)$ (i.e., the set of points $x\in X$ whose any neighborhood $O_x$ has positive measure $\mu(O_x)$) has countable cellularity and hence is separable.
Each $\sigma$-additive Borel measure on a Polish (= separable completely metrizable space) is Radon. This follows from the existence of metrizable compactifications of Polish spaces.