It isn't wrong (at least, if done correctly). I have no problem with using centrifictional force.

However, it is not simpler. It is unnecessarily more complicated. Errors tend to creep in. That's why I disagree with your contention in the OP when you said that the BA's explanation was "unnecessarily complicated." As I said before, changing reference frames is valid--but ignoring an effect is not.

PS: Part of the reason for ignoring the cetrifictional force, in this case, is that if you actually compute the force for the Earth, the amount and direction of the force is the same for every point on the Earth. There is no difference. Which means that the tidal force--which is a derived force much like centrifictional force is a derived force--has no contribution from centrifictional force. The tide-raising force varies in direction from one side of the Earth to the other.