According to all the artwork I have come across be it illustrations or video games, the majority of chainmail appears to be sleeveless with an additional "collar" worn on the shoulders or with short sleeves. No wrist length sleeves, integrated "gloves" or coifs. No mails chausses either. I know a lot of things can seems self-evident to us in retrospect (the chausses for example). Nevertheless I find at least the full sleeves/enclosed hands and head coif to be within a reasonable range of possible innovation. Maybe the coif wasn't needed because if they wore chainmail they logically already possessed a helmet. I would very much like to hear your informed opinions on this topic
Sincerely yours,
Koyraboro.
Its difficult to answer a question about why something wasn't done in the past even if, theoretically, it could have been done. Since "Celtic" armourers had the ability to make coats/shirts of mail they could have theoretically made anything out of mail; coifs, chausses, handbags, anything really.
The reason why their armour was made to protect the torso and the upper arms only is probably to do with the difficulty and cost involved in making such a garment. as you know, mail is made by interlinking rings which, in order to perform as armour, need to be individually riveted shut. Society in Iron Age Europe North of the Alps did not have the large polities of the era of the Crusades, which could support larger and more complex craft specialisation, making the production of mail garments an easier task - If you want sleeves, you need more metal and more time, same with chausses. Even in the medieval period when full-sleeved mail shirts and chausses were used, it appears that it was only available to some people. In the early 12th century Guibert of Nogent described a French host, of which the most beautiful were the "knights clad in mail armour and helmets". likewise, depictions of men in armour (such as the Northern Facade of the Cathedral of Notre Dame in Reims c.1220) Show varying levels of 'coverage' - some have coif, hauberk, and chausses, others have just the hauberk and coif without the chausses. Again, this was possibly due to prohibitive costs. We start to see from the mid 13th century onward more references to lower ranking individuals wearing armour, such as in the 1252 Assize of arms of Henry III of England.
So, while we can't say for definite why Iron Age people didn't construct full-coverage mail garments, it was likely down to cost combined with ease and speed of manufacture. There may have been other factors, such as weight - perhaps it was the general consensus amongst Iron Age people that slightly less arm protection was worth the trade off in speed and having a less tired sword-arm.
Edit: Sources.
Dowen, K. 2017. The Introduction and Development of Plate Armour in Medieval Western Europe c. 1250-1350. Fasciculi Archaeologiae Historicae. Vol. 30, pp. 19-28.
Feugere, M. 2002. Les armes des Romaines. Editions Errance
Brunaux, J. L. 2004. Guerre et Religion en Gaule. Editions Errance