You’re missing the part about the Qur’an actually being the miracle. Muhammad’s miracle.
OK, so this nicely dodges everything I’ve just said and redirects the discussion to something else!

But no problem, I’ll play along!
I certainly am not missing the part about the Qur’an being the miracle! When I read the Qur’an the first thing that struck me was how much of it is a critique of the Bible. In fact, having read the Qur’an, this idea that it is a miracle and serves as its own proof for divine authority is the strangest and one of the most self serving bits of Islam as far as I can see! A completely circular argument. Just saying something is a divine miracle doesn’t make it so. The Qur’an is a book ultimately, and it seems beyond me that God would choose to give the ultimate revelation of His divine will in a book in Arabic!
There are many arguments for the miraculous nature of the Qur’an, however moist Islamic scholars put the most emphasis on the first one—the unique literary style of the Qur’an.
By revelation Mohammed claimed that:
This Qur’an is not such As can be produced by other than God
10:37
He boasts that:
if the whole of mankind and Jinn Were to gather together To produce the like Of this Qur’an, they Could not produce The like thereof, even if They backed up each other With help and support
17:88; cf. 2:118, 151, 253; 3:108; 28:87-87.
It seems to me that eloquence is a highly questionable test for divine inspiration. Lets look at someone else amazing: Wolfgang Amadeus Mozart. Know one could question his eloquence, he composed his first symphony at the age of six, and produced his entire music corpus by 35. By comparison, Mohammed was 40 when he started the Qur’an. But what Muslim would say Mozart’s works are miraculous like the Qur’an? If eloquence were the test, then a case could be made for Homer’s works, Shakespeare could be a prophet too!
Many Arabic scholars agree that the Qur’an is not, in fact, unique in its literary style. C.G. Pfander for example, points out that:
it is by no means the opinion of of unprejudiced Arabic scholars that the literary style of the Qur’an is superior to that of all other books in the Arabic language…Some doubt whether in eloquence and poetry it surpasses the Mu’allaquat, or the Magamat or Hariri, though in Muslim lands few people are courageous enough to express such an opinion.
Pfander, C.G., The Mizanu’l Haqq (Austria: Light of Life, 1986), p. 264.
One Muslim who is courageous enough is the Iranian Shi’ite scholar Ali Dashti who points out that there are numerous grammatical errors and irregularities within the Qur’an, sentences which are incomplete and not fully intelligable without the aid of commentaries; foreign words, unfamiliar Arabic words and words used with other that the normal meaning; adjectives and verbs inflected without observance of the concord of gender and number; illogical and ungrammatically applied pronouns which some times have no referent; and predicates which in rhymed passages are often remote from the subjects. He lists numerous examples: 75:1; 4:160; 20:66; 2:172 and so on, one of which is verse 9 of sura 49 (ol-Hojorat) where God perportedly says:
If two parties of believers have started to fight each other, make peace between them
The verb meaning ‘have started to fight’ is in the plural, whereas it ought to be in the dual like its subject ‘two parties’. (cf. Dashti, A., Twenty Three Yeas: A Study of the Prophetic Career of Mohammed (London: George Allen Unwin, 1985), p. 48-49.
Anis A. Shorrosh is another author who catalogues many faults in the literary style of the Qur’an.
I could go on, but, the point I am trying to make is that even if it were the most eloquent book ever written in Arabic (which it’s not) it would no more prove its divine inspiration that a man’s strength would demonstrate his wisdom, or a woman’s beauty her virtue.
There is no logical connection between literary eloquence and divine authority.