One is that Chazal mistakenly believed the heart (together with the kidneys) to house the mind, and consequently, the soul. (I know that some people argued that the mind and soul are not necessarily seated in the same place. I think that they can quite definitively be proven wrong, but that is not the topic for now.) Chazal lived in a world where a person's mind and soul were mistakenly thought to relay their force and influence from the heart, via blood and breath, rather than from the brain, via neurons and nerves.
A second and distinct problem is that in Chazal's time, there was clearly no medical possibility of differentiating between the functioning of different organs and systems. Rabbi Breitowitz notes that in the ancient world there was no practical situation in which there was a differentiation between brain death and cardiac death. All vital systems - respiratory, circulatory, and neurological - would fail at approximately the same time, and there was no way of keeping one system going while another had failed. Thus, the failure of any one of them would be a satisfactory indicator of death. As such, any argument that the Gemara demonstrates Chazal to have conditioned death on a particular one of these systems is missing the point. As Dr. Noam Stadlan puts it:
The definition of life based on the presence of circulation achieved widespread acceptance both in halakhah and in the secular world at a time when the body could be considered an indivisible whole. This definition fails to yield logically cogent results in an age when the body is no longer seen and treated as an interdependent structure. It also conflicts with the halakhic definition of life that is applied in the cases of conjoined twins as well as decisions regarding transplantation.
Now, if you are of the view that Chazal's words on anything are sacrosanct, infallible and timeless, none of this is relevant. But if you are of that view, then you should also prohibit donating or receiving kidneys, which Chazal say provide counsel. Since pretty much nobody takes that approach, certainly not the RCA, then an analysis which does not take the aforementioned two factors into account is fundamentally flawed.