Very nice question! I think you already mentioned the main points and take the question to be about the function of each particular argument.
Regarding this, they are more or less well-caught in the SEP article on Aesthetic Judgement written by Nick Zangwill.
In a sense, his main argument is the second one. But without looking at the bigger picture, it is hardly intelligible.
To quote the question as stated by Kant himself:
How is a judgment possible which, merely from one's own feeling of pleasure in an object, independent of its concept, judges this pleasure, as attached to the representation of the same object in every other subject, a priori, i.e., without having to wait for the assent of others?” (§36, 288)
To quote yet another (and even more relevant) SEP article for where the argument is to be found:
The argument constituting Kant's official answer to this question (the “Deduction of Taste”) is presented in the section entitled “Deduction of Pure Aesthetic Judgments,” in particular in sections §§31–39, with the core of the argument given at §38. It is also prefigured in the “Analytic of the Beautiful,” in particular at §9 and §22, although the argument of §22, which appeals to the notion of a “common sense”, takes a somewhat different form from the presentation in the “Deduction of Taste” proper.
Function of disinterest
The argument of disinterest is to distinguish the subjective, yet not universal aesthetic judgements of personal taste from the subjective and universal judgements of beauty proper. Both involve aesthetic aspects and both involve pleasure, but they have to be distinguished. As Nick Zangwill puts it:
Moreover, unlike other sorts of intentional pleasures, pleasure in beauty is “disinterested”. This means, very roughly, that it is a pleasure that does not involve desire — pleasure in beauty is desire-free. That is, the pleasure is neither based on desire nor does it produce one by itself. In this respect, pleasure in beauty is unlike pleasure in the agreeable, unlike pleasure in what is good for me, and unlike pleasure in what is morally good. According to Kant, all such pleasures are “interested” — they are bound up with desire.
As such, the argument about disinterest is one about categorical differences and serves means of delimitation. One point I would like to clarify: Zangwill muddles the differences between intentional pleasures and moral judgements a bit here. Moral judgements are meant to abstract from intention which is "compensated", as it were, with producing non-intentional (disinterested) pleasure. This is why he has to delimit judgements of beauty from moral judgements later on.
If you like to read more about the link and differences between the moral and the beautiful in Kant - where the nature of the judgements of beauty is further determined - I suggest reading this related answer. (Disclaimer: It is an answer of mine)
Function of interplay of faculties
Here, it gets interesting. The interplay of faculties is a story that can be told which kind of suggests universality while it cannot really become a certainty. It is a demand, and necessarily so, the validity of which depends on many factors. Kant's main argument here is summarised in a footnote (§38, 5:290):
In order to be justiﬁed in laying claim to universal assent for judgments of
the aesthetic power of judgment resting merely on subjective grounds, it is
sufﬁcient to admit: 1) In all human beings, the subjective conditions of this
faculty, as far as the relation of the cognitive powers therein set into action
to a cognition in general is concerned, are the same, which must be true,
since otherwise human beings could not communicate their representations
and even cognition itself. 2) The judgment has taken into consideration solely
this relation (hence the formal condition of the power of judgment), and is
pure, i.e., mixed with neither concepts of the object nor with sensations as
determining grounds. If an error is made with regard to the latter, that
concerns only the incorrect application to a particular case of the authority
that a law gives us, by which the authority in general is not suspended.
Therefore, we are - stricly speaking - not justified in demanding agreement because of the interplay as such, but because we are justified in assuming that we share the same formal conditions, i.e. faculties in interplay and sensual input that are at work in the judgement. Not the least reason for being justified in this is because intersubjective communication about representations works, i.e. there has to be some common ground.
Objectivity and subjectivity
What the SEP articles actually fail to acknowledge or discuss the main argument and point regarding those "formal conditions" mentioned in §38 that Kant is developing up to the later sections (esp. §§74-78). As the discussion of Kant regarding the beautiful and the sublime in the aforementioned answer suggests, the harmonic interplay of faculties as such is not really the main argument, but rather a step, since a similar yet different harmonic interplay is at work in moral judgements as well!
For Kant, the beautiful, in producing a harmonic interplay between our faculties of imagination and understanding, suggests a purposefulness intrinsic to the (representation of the) object:
[N]atural beauty (the self-sufficient kind) carries with it a purposiveness in its form, through which the object seems as it were to be predetermined for our power of judgment, and thus constitutes an object of satisfaction in itself [...]. (Critique of the Power of Judgement, § 23, Ak. 5:245, Cambridge Edition p. 129)
This "objective" purposefulness of the object, as being grounded only in our particular perceptual equipment, is somehow "in between" objective and subjective: It is objective for every other rational being with the same perceptual faculties (see argument above), but subjective insofar it (even in ideal circumstances, see §33, §§74-78) only amounts for the beings with the same perceptual faculties (see esp. §77).
As this dips into one of the main arguments of the book and involves an understanding of what "reflective" judgements are, I suggest (re-)reading the introduction (or, even better, also the first introduction!), where he tries to highlight the line of argument more clearly than in the text itself. All I can do here for now is to give a hint to where this leads: Purposefulness as the subjective (in the very peculiar sense just described) maxim of the power of judgement that is shared by all humans.
As far as
the second maxim [i.e. "that of the power of judgment"] of the way of thinking is concerned, we are accustomed to calling someone limited (narrow-minded, in contrast to
broad-minded) whose talents do not sufﬁce for any great employment
(especially if it is intensive). But the issue here is not the faculty of
cognition, but the way of thinking needed to make a purposive use of
it, which, however small the scope and degree of a person’s natural
endowment may be, nevertheless reveals a man of a broad-minded
way of thinking if he sets himself apart from the subjective private
conditions of the judgment, within which so many others are as if
bracketed, and reﬂects on his own judgment from a universal standpoint (which he can only determine by putting himself into the standpoint of others). (Critique of the Power of Judgement, § 40, Ak. 5:295, Cambridge Edition p. 175)
In other words (providing the link to the question): In judgements of beauty, we reflectively (!) become aware of a purposefulness "from a universal standpoint", which really is rather a standpoint of shared subjectivity due to certain perceptual faculties, hence dependent on a (subjective in the sense of intersubjectively shared) maxim of the power of judgement we share with rational animals like us.
This means that because beauty proper is bare of our subjective (in the sense of individual) desires and preferences as it involves producing pleasure as intrinsic property of the object itself (linked to the subjective maxim of purposefulness of the power of judgement and the harmonic interplay between faculties that makes us necessarily assume purposefulness of the object), we validly claim judgements of beauty to be "universal" (for finite beings like us), justifying a demand to be agreed with.
This (i.e. purely object-induced production of pleasure without "subjective" input apart from the shared perceptual faculties) is basically the argument of Karl Ameriks in his relevant paper 'Kant and the objectivity of taste', The British Journal of Aesthetics, Volume 23, Issue 1, 1 January 1983, Pages 3–17, https://doi.org/10.1093/bjaesthetics/23.1.3. There, he also argues why disinterest is insufficient to delimit judgements of beauty from other types of judgement. See also later texts referenced in the second SEP article.
Thus, both arguments are but steps in the greater scheme of a) giving moral judgements, aesthetic judgements of taste, and aesthetic judgements of beauty, and judgements of empirical and philosophical knowledge (knowledge and science are discussed in §§78-83) their due place and delimiting them from one another and b) justifying purposefulness as the subjective, yet (in some sense) universal maxim of the power of judgement.
Remark: A lot of scholarly dispute (see esp. points 2.3 and 2.4) regarding the objectivity of judgements of beauty hinges on an understanding of what objective really means for Kant. Arguably, as Kant's objects of experience are always conceptually mediated representations, i.e. formed by our perceptual faculties, most misunderstandings arise out of an underlying (and - for Kant - wrong) assumption that objectivity has to mean "independent of our conceptual faculties". If that were true, there could not be objectivity in Kant. The waters are - admittedly - muddled by Kant's account of intuitive understanding as a possibility (§77), but this is arguably only a limiting concept that is supposed to negatively determine our understanding in making clear how it does not work like.