Primogeniture is simply not a thing, as in there's no succession laws to speak off : until the Late Middle Ages, it more or less simply "happens" with a huge bias in favour of precedents.
If the crown passed trough father to son regularily, it becomes more of a rule. If it didn't (as for English succession), there's more room for adaptations.
That said.
Most of the issues are coming from continental nobility.
It was attempted IOTL to have Arthur being raised in England by Richard, and was eventually rejected by Constance and bretons nobles, that had a really limited trust on Richard's benevolence (at the point Arthur refugeed in the Capetian court)
The main problem, eventually, was that Arthur didn't have much support in England or Aquitaine; and that he was too young to be a credible heir when his uncle John had an important clientele and the support of Alienor (and his deceased brother's eventually, on his last breath).
It would ask for Constance to die early ITTL and probably sees alliances switching with John acknowledging Capetian suzerainty over continental holdings.
Basically, and as always when it comes to Plantagenet's holdings in France, Capetians would have a decisive importance, and you'd end with a similar looking continuation of Plantagenet-Capetian wars, with the same actors but playing differently.
As for making Angevin hegemony looking like an unified entity, that's simply not possible.
Angevin Empire, in spite of its name, certainly wasn't an unified demesne.
It was rather a common feudal hegemony on really diverse demesnes, themselves often divided in small entities (especially Aquitaine, that was a true political mosaic).
Hence why the revolts of Henry II's sons fit remarkably the demesnes they recieved : Aquitaine, Anjou, Normandy, etc. each with their own identity, their own structures, their own interests.
All of that under the still present suzerainty of the French kings that could play on the feudal piano quite easily.
So let alone making it a single nation, just keeping the really various territories and nobility to simply get along is already hard enough.