Consistency implies $MSE(\hat{\theta}_n) \to 0$ as $n \to \infty$ for all distributions.

If $MSE(\hat{\theta}_n) \to 0$, then $\hat{\theta}_n$ is consistent.

Consistency and $MSE(\hat{\theta}_n) \to 0$ are equivalent definitions and require no additional assumptions.

Consistency is defined by the convergence of the bias, while MSE convergence is irrelevant to consistency.