The reason for thinking that consciousness is a physical process of the brain is the remarkable correspondence we find between injuries to the brain or the introduction of various chemicals, and variations of conscious experience. That leaves open the possibility that consciousness is a physically separate entity for which the brain is the interface through which it moves the body and receives sensation from it, rather than brain processes themselves being consciousness. However most of the ground for that is undercut by the fact that some brain injuries severing parts of the brain from each other also appear to sever the corresponding parts of consciousness -- split-brain observations. It seems that it is not only components of consciousness that correspond to brain regions, but also their interconnection. The more we find out, the less role is left for the hypothesis of a separate consciousness to do any work. Like the old "God in the gaps" argument to defend theism against science.

That is the argument, but it is important to note what it does not solve. It does not solve the problem of what consciousness is -- of why there is any such thing in the world as experience, and how any physical process could produce it. Nobody knows the answer to that. (ETA: including non-materialists. "There are souls" is not an explanation of how they work.) In this it differs from the problem of God. There may be people who claim a direct experience of God in the same way as they have direct experience of themselves, but it does not seem to be common. Experience of one's own presence, on the other hand, is reported by almost everyone. Yet everything else we know about the world tells us that there cannot possibly be any such thing. We cannot even see what an explanation for experience would be.

There are those who point to some physical phenomenon that is present whenever consciousness is, and conclude, "that's consciousness". Unless they show how whatever it is produces experience, they have not explained consciousness. Most proposed solutions are of this form.

There are those who point to some more or less speculative physical process (e.g. quantum gravity in the microtubules) and assert, "that's consciousness". Unless they show how whatever it is would produce experience, they have not even speculatively explained consciousness.

There are those who take the apparently impossible magnitude of the problem as an argument that there is no problem, which is like a student demanding full marks because the exam was too hard.

There are those who claim to have realised that in fact they have no consciousness and never did (Buddhist enlightenment is often so described, and I believe that the psychologist Susan Blackmore has said something like this, but I can't find a cite). If they carry on functioning like ordinary people then they are claiming to be p-zombies, and if they don't, they've philosophised a tamping iron into their brain.

As for myself, I simply note the evidence above, the problem it leaves unsolved, and my lack of any idea for a solution. Yet it seems that few people can do that. Instead, as soon as they start thinking about the problem, they frantically cast about for solutions and latch onto something of one of the above forms. The problem is like a piece of grit in an oyster, provoking it to encyst it in layer upon layer of baroque encrustations that merely hide the problem instead of solving it.

My point in this post is that I don't see why we have reason to reject the 3 possibilities above.

We have as much reason to reject them as we have to reject the existence of a slice of chocolate cake in the asteroid belt.

People in the comments have just taken it as a given that consciousness resides solely in the brain without explaining why they think this.

Yes, this is a regrettably frequent error on this question.