While not a Carolingian scholar by any measure, I'm well enough read to put my trust in the scholars who trust sources telling them Charlemagne was not literate. You are knowledgeable enough to start thinking critically about the man versus the myth, but this is an issue where understanding the less popular historical context is essential to not misconstruing the evidence. It is almost inconceivable to us that a man as powerful as Charlemagne who was responsible for initiating some of the most impactful reforms of medieval reforms in the first centruy could not derive some direct impact from his greatest non-martial achievements. I mean children can read let alone the ruler of much of Western Europe. Children are not, however, educated using Merovingian chancery script (p.24) (pdf - this is a very useful guide to scripts, albeit slightly confusin in layout). A chancery script was one used for essentially government business - it was a business script which was used for records. It is notoriously impossible to read but it was the script from which enabled the written aspects of government to function. Most people, including modern scholars, read this script with a finger tracing each word - outloud.

Did Charlemagne need to be literate to run an empire in the first millennium? No, there were numerous illiterate rulers in this millennium and the next - the social, political, and legal culture was primarily oral. Literacy was not a necessity.

Did Charlemagne know Latin and be sufficiently fluent to criticise others? Yes, this does not mean he read Latin but that he was taught it and understood it. Latin was not a dead language restricted to the 'educated'. You mention the educated individuals which Charlemagne surrounded himelf with, quite famously. Charlemagne listened to these, and less famous, individuals who would read Latin, debate, and provide a level of academic discourse unparalleled in medieval Europe at that time.

Your comments on Charlemagne's reforms is erroneous. He was not attempting to 'educate the denizens of his kingdom' he was instituting several centres of learning (most famously Aachen and Tours) which revitalised the ecclesiastical study but, as a healthy byproduct, led to the creation of Carolingian miniscule (p.45 of your pdf above) which is perhaps one of the clearest and most legible medieval hands, up to the Humanist handwriting of the fifteenth-cenutry (p.77). You'll probably recognise it (or the Humanist) if you used Microsoft Word before they switched to Calibri: it was the inspiration for Times New Roman.

Finally, there is the issue of what your sources mean by litteratus and what we understand by literate. On this my position is rather clearly set out in this response from some months ago. Yet there is little to indicate that Charlemagne practiced any of the features which would deem him to be literate by contemporary and none which even indicated functional literacy by modern standards. He might well have possessed functional literacy but the practices which surround the documents which survive do not lend themselves to proving such a supposition. I would enjoy a contribution from /u/bitparity, /u/depanneur, /u/Mediaevumed, and perhaps /u/textandtrowel to substantiate such an analysis.

it would seem this is incorrect, and merely a rumour, if not an outright fabrication (...) why might somebody wish for the King of the Franks to look like an uneducated man?

Sadly this is a projection. The circumstantial evidence provided in your argument (1. that Charlemagne could criticise Latin - reported by others; 2. that he was surrounded by literate and learned men; 3. that he advocated and patronised a series of schools) cannot support the claim that he was literate and much less that our sources are misleading or suspect. Instead I think this represents a projection of modern values onto the past. It is not a misconception but instead to the modern mind inconceivable that such a man as Charlemagne was illiterate. It doesn't mean he was stupid, or even uneducated. He was surrounded by some of the greatest minds of his age, and patronised some of the finest schools in Europe. If anything it should raise him in our estimations, but do not lose sight of his pragmatism and the dire necessity for such pragmatism in relation not just to learning but to the keeping of legible records in the Early Middle Ages. There is certainly no conspiracy among scholars to paint Charlemagne as illiterate - it would probably help no small number of theories is they could ascribe to him directly a single written promulgation or letter without accounting for the filter of others.

I hope that helps, if you want any studies to chase up let myself or pose the question to the wider sub'.