The covariant notation is concise and to some even elegant, but it is also less useful in practice. One often has to deal either with electric or magnetic field

The covariant notation is concise and to some even elegant, but it is also less useful in practice. One often has to deal either with electric or magnetic field specifically as opposed to EM field tensor, and even if one does not need that, it is often possible and useful to work with only electric (or magnetic) component in 3D vector notation.