Both of you are correct in this assertion. Many Ayyubid cavalrymen
were of Turkish origin. However, the assumption is being made that it naturally follows they wore Turkish armor. This is false. Only a minority of Ayyubid and Mamluke soldiers actually wore the lamellar armor they're often depicted wearing, as they are in this mod, during the 12th and 13th centuries. Furthermore, even Turkish armies south of Anatolia abandoned traditionally Turkish armor in favor of much more Arab-style armor. This can be seen in this source:
Notice how it says Mu'izz al Din "led a cavalry force armed with long lances and swords, wearing full-length mail hauberks and perhaps segmented helmets with plumes or crests." The Arab fashion they adopted is clearly much more reminiscent of the Crusaders themselves (and it should be noted that the Crusaders States gradually adopted more Arab, Byzantine, and other local styles, such as turbans, lightly colored coats over their armor, and lamellar). Also, the segmented helmets are also another example of Arab armor. The Arabs had used conical segmented helmets since the very early Medieval Ages, both nasal and non-nasal ones, such as these:

This second picture is actually of Nur ad-Din, the famous (Turkish) Zengid ruler. It's no coincidence he's wearing an Arab style helmet here rather than a Turkish one, because the Zengids were further south from Anatolia.
Hence, an actual typical Ayyubid soldier during this time period would've looked much more like this. He has the correct chain mail armor and helmet, as well as a much more realistic cloak over the armor, compared to the extremely over-stylized ones we tend to see with Western depictions of medieval Arab armies.
Speaking of over-stylized clothing, the main reason we see Islamic soldier depicted with those so often is more than likely because of this:
The Crusader chroniclers overemphasis of the Turcomen in Ayyubid armies obviously led to the stereotype in the Christian European mind that
this is how Arab and Muslim soldiers dress. Extremely light with tons of colors and designs. It didn't help that they mainly served in the auxiliary as well, meaning they would often be the first thing of the army that the Crusaders would see.
Also, I think its worth mentioning
why Arab armor would be so prevalent over Turkish armor in Ayyubid armies. This was mainly because of the Iqta' system, as described by esteemed Islamic historian Sato Tsugitaka here:
Similarly to the feudal systems of Western Europe, when the King would declare war, the muqta', who would typically be Emir's (Arabic for Lords), would have to gather the peasants and other men of the iqta' to lead them in war, while also helping supply and pay for their armor. Considering that most of the armor produced in the Dominions of the Ayyubid Sultanate in this time period was by Arabs, armors of the Arab style, as shown in the image of the soldier above, were much more mass produced compared to foreign produced Turkish ones in Anatolia. Considering that the Emir's would naturally try to keep the expenses for the armor for their men as low as possible, they would obviously purchase much more common, locally produced, and thus inexpensive Arab style armors rather than trying to import foreign Turkish armor, which would be more expensive. This is also why the Zengids used mainly Arab armor as well, as described in the sources and images above, as they adopted the Iqta' system from the Seljuk's and most of their trade would've been with the much more organized (following the Seljuk collapse) Arabs and Crusaders and their own blacksmiths would've been mostly Arab as well.
Also, I checked out some of the Nihayat al-Su'l and Psychopathy is correct in saying that "Muhammad ibn 'Isa al-Hanafi al-Aqsara'i makes it clear that Arab Egyptian and Arab Levantine soldiers wore mainly mail and plated mail in the 1200's." For the plated mail, it specifically talks about "jawshan" which essentially refers to armor such as this (which is only partially survives and is one of the few known Mamluke armors that still exist according to MET):
After saying all this, I want to make it clear that I'm not saying it would've been absolutely, 100%, and without exception, impossible for any soldier in any Ayyubid army to ever be wearing Turkish armor. Obviously, some did and their usage only increased in the 15th century as the Ottomans gained more and more influence. The issue is that the
vast majority of Ayyubid soldiers in this mod are wearing Turkish-style armor that is clearly historically out of place, especially in the High and Early Late Medieval Ages, while quite a large number of the units from the Kingdom of Jerusalem's roster do a better job of representing the Ayyubid army than the Ayyubid's own roster.