The problem wasn't that Henry didn't have a clear successor, it was that he had two.

He has made his court swear alligence to his daughter, Empress Matilda, but she was in Germany being an Empress when he died, so his nephew Stephen took the crown. The country fell into Anarchy because Stephen was a terrible king and could never really get rid of Matilda and her brother, Henry's iligitimate son.

But your question: why didn't they go back to being Saxons was because William the Conqueror was an excellent administrator. A better name for him would honestly be William the Actuary. He set up a vast network of middle managers, who are harder to dislodge than a fuedual overlord.

He is know for building a series of castles and earldoms of Norman and French allies all over the country, but his most long last contribution should be the Doomsday Book.

This was a massive census of every person, animal, farm, house and building for the purpose of taxing and organizing the whole country. His sherrifs (litererally: shire of the reeve) would organize the local population into something called "hundreds", a census term that was in use until 1894, and they would oversee land disputes, justice, weddings, births, deaths and taxes.

Instead of just being a group of fisherman who occasionally would get axe murdered by a Dane, now you were a village or hamlet with a name and municipal structure.

The Normans completely changed the face of the island. There really was no going back.