By definition, a consistent estimator or rather a weak consistent estimator is one that causes data points to converges to their true value as the number of data points increases. So naturally, bias which is defined as $||{E[\hat{\theta}]-\theta}||$ converges to $0$ as number of points $m \rightarrow \infty$. So, even variance is $0$ as $E[(\hat{\theta}-\theta)(\hat{\theta}-\theta)^{T}]$ will have $\hat{\theta}-\theta=0$.
However, I am still not too sure if I am thinking in the right direction due to the Kolmogorov SLLN as well as the no-free-lunch-theorem.
Just another question, since an unbiased estimator has bias=0 and hence, the points are all at their true places in the empirical distribution, then is it not the definition of "strong" consistency?
Consistency does not imply that variance goes to 0. You can find counterexamples for this:
https://stats.stackexchange.com/questions/74047/why-dont-asymptotically-consistent-estimators-have-zero-variance-at-infinity