The question is put in many equivalent ways: is thought an effect of the brain? is the brain responsible for thought? are all mental states brain states?
The question doesn’t strike me as much different than asking “is the hand the organ of grasping?” or “is the leg responsible for kicking?” or “are all digestive states stomach states?” The answer is certainly yes, and to resist entirely the force of the answer seems largely to argue for the sake of argument. It is equally evident that all of these actions and states are said in the mode of an instrument. The hand certainly grasps, and the leg certainly kicks, but not in such a way as we conclude that the human being does not grasp or kick anything. In fact, it is precisely because a human being kicks something or digests something that a leg or a stomach does something. The actions of the part are caused ultimately by a principle agent, even though they might be proximately caused by another part, like a hand is moved ultimately by a human being, even if it is proximately moved by the tendons, and remotely by the brain.
Many of the discussions about the brain’s causality of thought seem to involve the idea that if one makes the brain responsible for thought, that somehow it becomes the principle agent of thought. This is as silly as thinking that if one makes the hand responsible for grasping, that somehow it is the principle agent of grasping, as opposed to an instrument used by a human being. I call this conception silly because it is sees the brain in exactly the same way as the arm is portrayed in Dr. Strangelove.
A separate difficulty involved here is the many meanings of the word “thought”. The word can apply to the activity of the interior senses as well as to intellect. If the question of “does the brain think” is phrased in such a way as to reduce all thought to sensation, that is another question.