OTL, King Henry II of England ruled a vast Angevin Empire which consisted of lands in England and France.
That alone needs to be greatly nuanced. The "Angevine Empire" was a collection of diverse political entities, some relativly unified as Normandy and England, the others largely divided as Aquitaine, and each with their own customs and sometimes institutions.
While he was at the head of this demesnes, saying he ruled them equally and entirely isn't reflecting the historical situation, and why the tentatives of Henry II to go against these relatively coherent entities were unsuccessful.
He suffered a great revolt against his rule by his traitorous sons; Young Henry, Richard, and Geoffrey.
You overlooked that the Great Revolt was caused not because the sons and enemies of Henry II were mischievous, traitorous and opportunistic; but because he actually tried to give John Lackland...some lands.
His other sons didn't agree on this, mostly because they saw this as a dismembrement of their respective demesnes (Aquitaine, Anjou, Normandy), being supported by their respective nobility that saw this as an important political intervention and therefore a threat to their own power.
King Henry crushed this revolt, called the Great Revolt, and apparently forgave his rebellious sons.
Crushed? Not exactly. Henry II dealt with outer threats, especially Scotland and France, but didn't really advanced against his children, and the peace of 1174 was more of a compromise about status quo than an imposed peace.
What if he had been more ruthless in the aftermath of the first revolt, as he should have been such as excommunicating his rebellious sons and bequeathing his ENTIRE realm to his loyal son, John?
More ruthless would have meant no peace in 1174. Remember that we're not only talking about rebellious sons, but rebellious lords and territories.
Could Henry II have gone trough a full scale campaign in Western France against Normandy, Anjou, Brittany, Aquitaine and most probably France and Flanders, with no real reason for his sons to make peace?
Politically, strategically and along contemporary mentalities, it wouldn't have make sense.
Would the Angevin Empire continue for a lot longer instead of losing its French possessions only a few decades later?
If something, it would have made it falling apart earlier. Remember, we're talking of an hegemony over really distinct entities, not a real unified empire there.
With a more ruthless Henry II, unwilling to compromise, he would have certainly damaged more his domination.
(Not to mention John Lackland lack of real competence as well, as pointed out with his actions in Ireland)