Basing off this article, the quoted sources, and other Chassidic sources, the truth is that everything you describe in the question can only account for the physical functioning of the body. However, you are not acknowledging the experiential functioning of consciousness as being a separate thing.
This is not clear at all. It may be accepted, a priori, in academia nowadays that the solution to the mind-body problem is that everything is just body, and consciousness is heuristic patterns upon patterns that we simply have to figure out. However, this has not been proven or demonstrated yet, and according to Torah sources (see the article, or the quote below), as well as well established philosophical sources1,2, consciousness is something that cannot be explained through materialism.
A computer can mimic a human's abilities, such as reasoning, and out-perform the human on them, but the computer can't have a conscious experience of "what it is like" to reason. The concept of qualia, which is non-physical information that is not based on any logical principles, is not something we can re-create, or even have any in-roads at the moment to creating (I am not aware of any active fields of research to create a new colour, or program an AI to consciously experience colour for example), and therefore no matter how well we can describe every last mechanism within the human body, we can never explain why it is we feel pain, or any other sensation the way we do. We know that this chemical breaks down into that, which causes this cell releases that neurotransmitter and this leads to x which leads to y, but at some point z leads to a step which we cannot ever hope to bridge using reasoning and logic: what the colour red looks like.
This is where the soul comes in, this is where Hashem's immanence, being the One Who can do absolutely Anything, including bringing forth conscious experience and qualia from absolute nothing (not even logical principles), is required to explain the matter.
This is not the entirety of what the soul is or does, but this should be enough to answer the question. I'll leave off with a quote from the Lubavitcher Rebbe, Sefer Ma'amarim 5730 p. 143:
דאמיתית המציאות של היש הנברא דוקא הוא היש האמיתי. כי דוקא ביש הנברא
ישנה ההרגשה שמציותו ואינו עלול מאיזו סיבה. שהרגש זה שרשו רק בעצמותו
ומהותו יתברך שאינו עלול מאיזו עילה שקדמה לו...
The essence of the created self is the Divine Self. For only the created self possesses a consciousness that its own existence is self-defined and has no cause or source. This consciousness can derive only from the very essence of Hashem, may He be blessed, Who has no cause or source that proceeded Him...
In context, the "created self" is referring to the soul.
With regards to your point about whether the soul reaches all the way down to the physical, Rabbi Shneur Zalman Schneerson writes in Likutei Amarim—Tanya, Shaar Hayikhud Ve-ha-emunah, Chapter 1.
“even in the literally inanimate—like stones, dust and water—there is
a soul and spiritual life… which enlivens and creates the inanimate,
that it may emerge as an
existence from nothing…”
The article I linked at the start goes into this a little, and you can find sources there. "All reality is Divine thought thinking Itself". This is a clear distinction between the philosophy of Amalek, and Torah. Amalek claims that if you go down to the core of the physical, you'll find it is dead. Just dead atoms following dead laws.
Torah, on the other hand states that the core of reality is Hashem's Essence, the Living God Himself. It is alive. What is the life of a rock? Remember, according to Zohar, a soul, essentially, is a mission, a purpose. The rock's soul is it's purpose. Every rega, the Living God personally, purposefully, recreates the rock from absolutely nothing. What does He want for the rock? That it should not be destroyed unless for a positive purpose to aid man. Hashem cares about that purpose infinitely, He's paying infinite attention to that rock, and if the rock is used for a mitzva, or the opposite, His Ratzon (the highest source in creation of created life) is fulfilled (or the opposite, God forbid).
There is so much more to say, this is not a topic for a short answer, or beginners, and I pray that my words are clear and correct. Just take away this: the core of reality is alive, not inanimate, not dead.
1 - https://plato.stanford.edu/entries/qualia-knowledge/
2 - Of interest: The Emperor's New Mind: Concerning Computers, Minds and the Laws of Physics by Roger Penrose, although he suspects that the answer can be figured out within physical systems.