That's just the simplified version, jura!
The 'English' were a pretty mongrel bunch by 1066 anyway as they were a mix of Celtic (the 'original' Iron Age population), whatever blood came in with the Roman legionaries, who came from all over the Roman Empire, Germanic (and, I think, Gaulish) Saxons and Scandinavian Danes. And that, too, is a simplified version...
As I understand it the worst thing the Normans did was to introduce the Salic Law of inheritance (male line only) whereas women had been pretty much equal with men for inheritance in pre-Norman society.
I saw the tapestry many years ago at Bayeux. It was very nicely displayed but there were hordes of people there making it annoyingly slow to reach and then to view (because there was always someone obscuring the bit you really wanted to look at..). I thought the French explication of it was a tad triumphalist...