What seems to be omitted is that Aristotle was a realist about forms. He does call them secondary substances whose existence depends on the existence of primary substances (particulars, e.g. material things), but he does say that both exist in re, see Aristotle's Theory of Abstraction by Bäck, Ch.2. Furthermore, the "structure" is not a mere epiphenomenon of material interactions, as understood today (Averroes called something like that a "material" form, as opposed to "substantial" one), matter itself is only potential, it is form that brings potential into actual in what Aristotle calls entelechy (roughly, embodiment, actualization).
Dependent existence means, and this is a difference with Plato, that forms do not float separately from their particulars, which are the unities of matter and form, however, transplanted into a receptive medium, they could be separated. Normally, the receptive medium is human intellect (Greek nous) that abstracts forms from their material instantiations. As for the soul specifically, Cohoe investigated the issue in detail in his recent thesis The Human Intellect: Aristotle's Conception of Nous in his De Anima, who concludes the following:
I consider whether the separability of intellectual activity from the
body would be compatible with Aristotle’s overall views on the soul and its relation to the body, particularly his claim that the soul is the form of the body. I present two alternative interpretations of Aristotle’s conception of the soul: (1) the soul is constituted by a unified and interrelated set of powers; (2) the soul is ontologically prior to its powers and is not constituted by them. I conclude that both interpretations are compatible with the relevant texts... If the intellectual power and its activities can exist separately, then when, after the destruction of the human body, they do exist separately, the human being also continues to exist. I argue that this view is preferable for textual and philosophical reasons to a position according to which the primary subject of understanding is the intellect or the soul or one according to which understanding switches primary subjects after death.
"Where" the soul might exist is a curious question, but Aristotle did have Divine Nous. Even on the first reading there might perhaps be a way to make Aristotle into something like a functionalist (with a good deal of stretching, in my opinion). Goldberg tries it in Is Aristotle's Philosophy of Mind Functionalist?, but even he calls him an "antireductionist physicalist", as "the only ontology that Aristotle countenances, at least in the sublunary realm, is one of ordinary matter. Nonetheless souls, like syllables, are not reducible just to that matter; they are matter in particular forms." But even Davidson's nonreductionist anomalous monism (no mental substance but irreducible mental properties) does not make forms as much in re as Aristotle does.
Aristotle's obscurity on the soul led to diverging interpretations during the middle ages, first by Islamic and then by Christian philosophers, to whom the theological concerns made the issue paramount. The two most inluential ones were fashioned by Avicenna, who augmented Aristotle with neo-Platonic elements borrowed from Plotinus, and Averroes, whose conception influenced Aquinas. Although Aquinas, unlike Aristotle or Duns Scotus, was not a realist about universals he was most certainly a realist about souls as substantial forms. But he added a special twist that anticipated Kant and distingushed him from all other scholastics, see Aquinas: Body and Soul on SEP.
Aquinas did not consider existence to be a property, part of essence, but characterized it rather as an act, esse. While a soul without a body is incomplete in essence, as Aristotle held, it can still subsist through its immaterial esse, and the body itself receives its existence through the same esse when it is insouled. This allowed Aquinas to maintain the tight Aristotelian link between body and soul in his largely empiricist theory of cognition, but at the same time allow for survival of the soul upon destruction of the body through esse, as the Scripture demanded. This survival is however deficient, soul is only "subsistent" through incorporeal activities like thinking and willing (interestingly, animal souls are not), but it is not a "substance complete in essence" (although God might perhaps convert it into such an "angelic" substance after the resurrection). The last two chapters of Kenny's book Aquinas on Mind give a detailed account of Aquinas's view of the nature of the soul and its relation to the body.
As for observable effects, there is no problem with soul affecting matter, in Aquinas's metaphysics the higher can always affect the lower, but not vice versa. So soul can act on inert matter but not vice versa, see Aquinas and the Content Fallacy. Aquinas faced the opposite problem for his empiricism, however: how could matter (the lower) affect the intellect (the higher) in perception and cognition, even going through senses. This he solved by appropriating an entity barely mentioned by Aristotle and made into a part of the Divine Nous by Avicenna, and making it a part of human soul, see What is the agent intellect according to Avicenna and Aquinas?:
"It is necessary to postulate a power, belonging to the intellect, to create actually thinkable objects by abstracting ideas from their material conditions. That is why we need to postulate an agent intellect."
Instead of "receiving" material forms through senses, which would require matter acting on intellect, the agent intellect acts the other way, by actively processing what is received through the senses (stripping forms of their material conditions). This was a novel idea, also anticipating Kant with his active cognitive subject. Let me finally mention that Kant modeled his sensation/concept unity in perception on the Aristotelian matter/form unity in re.
