Both the Bayesian perspective and the optimization perspectice are legitimate ways of understanding the Kalman filter. I like the Bayesian perspective better.
Forgive me, I'm thoroughly confused by that dichotomy. How are they different? Approaching from bayes rule or a "maximum likelihood" approach produces the same results.
The result is identical, the understanding is different. I would suggest that the Bayesian perspective leads to insights like the UKF [1] which IME is all round much better than the apparently better known EKF for approximating non linear systems.
[1] That is, it is generally easier to approximate a distribution than a non linear function.
The problems of the filter are present in both.