There was no single Anglo-Saxon Language before the Norman Invasion. Baugh & Cable, (A History of the English Language , 4th edn, 1993) tell us that there were four dialects. The two north of the Thames, Northumbrian and Mercian had enough similarities to be collectively referred to by some as Anglian. The two southern dialects were West Saxon and Kentish.
By the time of the Norman invasion, these dialects were very different from what they had been four centuries before. This is hardly surprising. Very few people indeed could read or write, and it is the written word that tends to slow down change in language, a change in the dialects that would become English that had been accelerated by the influence of Viking invasions and of church Latin.
The English dialects after the Norman invasion were largely spoken by the peasants and serfs, Norman-French and Latin being the languages of the ruling class and the majority of literate people.
By the time English began to become the language of all classes in the middle ages, the influence of Norman-French had made a considerable difference to both the grammar and vocabulary of the previous largely Germanic language.
Whilst the language of Chaucer was clearly more like modern English than was, for example, the West Saxon of four centuries before, it is not really true that "essentially understandable" for the modern reader. I doubt if many people today could understand the first four lines of the Canterbury Tales that you quoted. Some of us can, because we studied it at school or university, but that does not mean that everybody can. This would be even truer of the spoken language, which would sound like a foreign language to most native speakers of modern English.
Even the language of Shakespear, two hundred years after Chaucer and a mere four centuries ago is not easy, as these lines from Macbeth show: If it were done, when 'tis done, then 'twer well,it were done quickly: If th' Assassination could trammell vp the Consequence, and catch with his surcease, Successe: that but this blow might be the be all, and the end all. Heere, but heere, vpon this Banke and Schoole of time, wee'ld iumpe the life to come. But in these Cases, we still haue iudgement heere, that we but teach bloody Instructions, which being taught, returne to plague th' Inuenter.
A person who had not studied Shakespeare at school, or seen a play would find that fairly difficult.
The spread of the printed word after Caxton's introduction of the printing press towards the end of the fifteenth century did help fix, to an extent, the spelling and grammar of the language. The approach to uniformity in the language was also helped by the stability, compared to earlier centuries, of central government, the absence of foreign invasion and, by the end of the nineteenth century, the introduction of universal education in Britain.
So, it may be that, in the four centuries between 1000 and 1400 the language changed more than it did between 1600 and 2000, but the reasons are reasonably clear. I think it is also clear that the earlier change was not so drastic, nor the second so slight, as you suggested in your question.