The brainstem also happens to have the greatest diversity of kinds of neurons, and is the oldest in terms of evolutionary biology. This lends to the idea that all creatures are conscious but humans are conscious of more things (via the cortex). Basically the cortex expands consciousness but the brainstorm is the source of it.
It's also worth noting that modern LLMs architecture are a lot closer in kind to the cortex and are dramatically different (simpler) than the brainstem. This can maybe afford some optimism that AI may be more likely to extend human consciousness than replace it, at least for a long while. (Even if it's not evenly distributed...)
I believe that their hypothesis is very plausible, because not only it has good experimental support from the effects of brain injuries and of anesthesia, but it also seems more likely based on evolutionary history.
In the beginning, in the most primitive vertebrates, the most anterior parts of the brain, the precursors of the human cerebral hemispheres and thalamus, had a much more restricted functionality than they have now in mammals, they were just extensions of the neural tube where the primary processing of the information from smell and sight was done. At that time, smell and sight were the only long-range senses and the only 2 senses that required a complex processing (the original function of the ears was only for sensing acceleration and angular velocity, hearing evolved only later).
After vertebrates evolved towards greater sizes and more and more complex structures, the brain also became bigger and added many new functions. Which parts of the brain have grown differs between the various classes of vertebrates. In mammals, most growth happened in the cerebral hemispheres, so most new functions are located in them. In mammals also the cerebellum has grown much in comparison with other classes.
In the first vertebrates, the neural tube had a somewhat similar structure from the spinal cord in the back until the midbrain in the front, with sensory nerves, motor nerves and neural networks interposed between those. In the front of the midbrain, the precursors of the cerebral hemispheres and thalamus had only a sensory function, for smell and sight. So at that time the likely location for the consciousness was at the anterior end of the neural tube, in what will became later the tegmentum of the brainstem.
The dominant theory today is that when in mammals the cerebral hemispheres became bigger and bigger, implementing more and more new functions for processing the sensory information from the entire body, for memory and for the control of the motor actions, the location of the consciousness also migrated forwards, from its old location from the brainstem to some place in the cerebral hemispheres.
Nonetheless, there is no obvious reason why would have been any need for the location of the consciousness to change, instead of only implementing new functions in the cerebral hemispheres, most of which work subconsciously, with only a superficial control from the consciousness.
The hypothesis from TFA is just that during the evolution of the brain of the mammals, until humans, the location of the consciousness never migrated out of the tegmentum of the brainstem, but it just remained there in its original position.
The only argument in support of a migration is that after the hemispheres have grown so much, the brainstem has remained very small in comparison with them, and one would have expected a more luxurious residence for the consciousness.
Nonetheless, there have been many cases of various kinds of damage to the cerebral hemispheres, and while that lead to severe cognitive impairments, the person remained perfectly conscious. In contrast, injuries to the tegmentum of the brainstem always result in coma.
In reality, the conscious thinking does not do much work in comparison with the huge amount of subconscious work that must be done for even simple things like recalling an old memory or recognizing some image or performing some acrobatic movement. The conscious thinking is more like an agentic harness, while the subconscious work is like what a very big LLM does.
So it is plausible that even a relatively small volume, like that from the tegmentum of the brainstem, might be sufficient for consciousness, while the huge cerebral hemispheres are necessary for doing all subconscious work and storing most of the memory.