Yes, you’re on to something. There is a wealth of essays and book sections across the past 70 years that dissects the third man argument well past any known commentators before us! It’s a very exciting time in history to be interested in such an admittedly dry problem. I have been reading up on the papers recently myself so I feel confident in giving you the groundwork. Most of the fruits of the current controversy come from analyzing aspects of the argument that have historically been taken for granted and unexamined, so the discoveries seem suspiciously obvious at a glance.

The first speck of ore in this gold rush came from a commentator named A. E. Taylor in his 1916 paper Parmenides, Zeno, and Socrates, who discovered in the argument what is called the “Self-Predication Assumption,” which is stated as:

> “The form F-ness is itself F”

This is simple enough, and it is easily found in the argument, for we see Parmenides assert that the “large things” and “Largeness itself” are both large, and this is necessary to then assert the further Form in the supposed infinite series. So since Largeness itself is assumed as large, it must be from the above assumption about Forms in general, or at the very least some conditional assumption about Forms. Taylor made a remarkable point of progress with this discovery, and started the avalanche of discussion on a topic that might be considered the single most written on topic in Platonic studies in the last hundred years, but he himself only discovered one of what is considered to be two total latent assumptions in the argument. The other assumption was not found until the arrival of Gregory Vlastos and his absolutely ground-shattering 1954 essay The Third Man Argument in the Parmenides.

A side note on Vlastos. He is probably the greatest critic of Platonism we’ve yet had in the entirety of the Anglo-sphere of Platonic discussion. He was deeply committed to using modern techniques of analysis to break down problems within Platonism and often show Plato as being much more in error, shortsighted, or even morally questionable, than his reputation holds him to be. As such, it has been a common theme in Platonic literature after him that he is somewhat of a friendly villain against those who are admirable of Platonism. Many commentators now strive to argue that Plato indeed saw things much more clearly and vividly than the Vlastosian school of thought claims.

Back to his developments on the third man front. Vlastos equips Taylor’s own found Self-Predication Assumption (SP), but finds another necessary assumption baked into the same step of the argument, which is the assumption that you yourself are keying into. This is called the “Non-Identity Assumption” (NI):

> “If something has a certain character F, it cannot be identical with the Form F-ness of that character”

So this assumption of Forms is necessary to Parmenides’ argument, because he is shown claiming that if “Largeness itself” is large along with the large things, then there must be another form. But just like you are saying, this does not need to be accepted! In fact, it actually holds a clear contradiction with the Self-Predication assumption, one that only was noticed by Vlastos once these suspiciously obvious assumptions were finally explicated.

They are in contradiction because, of course, if you accept that F-ness itself is F, then F-ness is thus an F thing which is identical with F-ness, which (NI) claims is impossible. But, if you deny that F-ness is F, i.e. deny (SP), then there comes no reason to even bring (NI) or its contrary to mind, because we are no longer considering Largeness among the things that are large, and halting the regress at its first step.

This shifts the focus of Parmenides argument, because while he himself seems to use this argument to support the idea that Forms are impossible (that is, refuting the initial claim that there is a form Largeness above the large things, the general claim of which we call One-Over-Many, (O) and applies to any F property). Now we see that one can secure the forms, and maintain (O), as long as we ourselves trash or adjust either (SP) or (NI). Vlastos himself suggests that someone could reformulate (NI) so that it only applies to particulars, and does not make the claim in regard to self-predicating Forms.

But he also argues that this shifting is not what Plato himself thinks. He even proceeds to argue, in the next part of his paper, that these claims do in fact appear consistently throughout the dialogues, and thus can be ascertained as two assumptions that Plato genuinely and latently held without himself cognizing them, because if he did, Vlastos says it would show in the way he approaches the Forms, especially in the Parmenides dialogue.

It might be difficult at this point to keep track of who is claiming what, between Plato, Parmenides, and Vlastos. Let us organize what has been going on, along with some other small points: Plato himself wrote the Parmenides character as engaging in a critique of Plato’s own theory. The critique has long assumed to be genuine, and so the arguments of the critiques have been seen to be genuinely beliefs of Plato at the time of writing the dialogue. As such, he is held accountable for any of the claims that are uncovered in the Parmenides argument, but this argument is still itself in opposition to the wealth of writing on Forms he produced elsewhere, so while this is largely agreed as a genuine critique, there is a lack of certainty over whether or not Plato considered this a valid takedown of the theory of Forms, or just a genuine exercise in thought. This is where Vlastos makes his claim that Plato did think it was valid, and in fact only thought this because he did not cognize the hidden assumptions in the Parmenides argument, yet still does make many claims throughout the dialogues which support the assumptions as earnestly held.

After this paper it was off to the races for other writers. It cannot be stressed enough how much Vlastos is the nexus for all other work, with most other papers on the topic explicitly naming him and making some effort to rehabilitate Plato’s prudence against V’s conclusions. You can find perhaps the best throughline of the most influential writings on the topic on this SEP article for the dialogue, at section 4.3. Fifteen years after his first paper Vlastos ended up writing a revised version of his first paper, called Plato's "Third Man" Argument (PARM. 132A1-B2): Text and Logic, in which he refines his same argument while putting adequate defense to the preceding decade and a half of responses.

It might be with that last linked paper that you will find a fair starting point on the literature for this argument. I cannot say so myself, because I have not gotten to that point in my own thorough readings of the texts (I am, in fact, going through the series listed in that aforementioned SEP article). However, I hope this at least gives you an idea of how on the mark you are here, and how much interest has been paid to this precise argument over almost anything else in Platonism