Does this black hole magnetohydrodynamics equation even superficially make sense?
I believe your assumption of the partial derivatives with respect to time being identically zero is incorrect.  When you are talking about working in phase space, i.e., Liouville's theorem with functions of positions and velocities, then you are correct that the partial derivatives of one phase space variable with respect to the others is zero.  But here (I presume), they're looking at the temporal dynamics of the black hole magnetosphere.

It is a little dense (but if you're working through Equation 4.24 above then you'll have no trouble with this), but you might find the stuff in this paper as filling in a lot of the background equations.