The affinity argument of Socrates is to support the immortality thesis of the soul. The argument is achieved by an analogical reasoning. The world of the Form and the soul share similar characteristics. The world of the Form is divine, intelligible, non-composite, unchanging and immortal. The soul is also divine, intelligible, non-composite and unchanging. Thus, the soul must be also immortal.
To evaluate the argument, you need to examine whether the analogical reasoning holds. If you can show that the analogy fails, then you show the unsoundness of the argument. There are at least three ways to attack the argument. (Some ideas borrowed from https://plato.stanford.edu/entries/ancient-soul/)
The argument assumes that there are only two kinds of worlds relating to existence (your first premise): the world composed of mutable things, and the world composed of immutable things (Forms). But it is possible to think of a world that shares some properties of the world of the Form, but not all. Some philosophers of mathematicians, for example, believe that mathematical entities are intelligible and immutable, but their mode of existence is different from that of the Form (e.g., structuralism and Quine-Putnam indispensability argument for mathematical realism). That is, it is possible that there are more than two worlds relating to existence, and that the soul belongs to a third world. In this case, the fact that the soul shares some properties of the Form does not warrant the conclusion that the soul shares all the properties of the Form, specifically, the property of immortality.
- Psychophysical reductionism
The argument also assumes that a person is composed of the body and the soul (your premise 2). But many materialists believe that all mental states, including the soul whatever that is made of or does, are reduced to the brain states. To them, the soul dies when the body dies, just as a smartphone dies when it is drowned.
- Underdetermination of probabilistic reasoning
This objection utilizes your complaint that a probabilistic argument does not prove much. The affinity argument only asserts the akiness in properties between the soul and the Form. Since all Socrates showed is that the soul is more like objects in the world of the Form, it is still possible that the soul after all does not belong to the world of the Form. Thanks to the aforementioned properties, the soul can be placed higher than the body, but lower than the Form. In other words, it can be placed inbetween mortal existence and immortal existence, possessing a demigod-like existence. So, the soul might have a longer life than the body, but would die eventually. This objection is essentially Cebes' weaver argument (87a-88b). According to the weaver argument, the soul in the body is like a man (weaver) wearing a cloak (weaver:cloak::soul:body). The weaver outlasts many cloaks, but he doesn't outlast all cloaks. Similarly, even if the soul survives the deaths of some bodies, this doesn't mean that it survives the deaths of all bodies.