And that, as far as I understand, tended to come down to which side the sub-vassals took. If Richard/John/Geoffrey's vassals in Normandy, Anjou, and Aquitaine backed him against Philip, he's in a pretty strong position. If, on the other hand, a lot of them take their King's side over their Duke (as happened IOTL), then things are likely to go Philip's way.
My question is why they'd tend to take one side over the other. John's high taxes and ham-fisted political style was certainly a major factor. I'm wondering what the other major factors there would have been if we replace John with someone more competent.
Yes. Since mediaeval kings , by and large, had no standing armies of their own, they relied for military force on calling out their vassals. That was the very nature of feudalism.
So Phillip of France called out his vassals to send knights to aid him to force his rebellious vassal (the Duke of Normandy, aka King of England) into submission. Not responding to such a summons was itself rebellion.
When a vassal received such a summons he in turn summoned his sub-vassals, to make up the numbers he had to send (because he didn't have a standing army either). In theory, and law, the Duke of Normandy's English vassals (those who held lands from him as King of England) did not get involved at all. Certainly John could not compel his English subjects to assist him in France. (There is an absolutely fascinating exception to this, which awaits a modern day Maitland to explore, which relates to those in the North of England, who held by drengage - the whole question of tenures by drengage and thegnage is most unjustly neglected, but that is for another day). However, there were always ambitious men willing to voluntarily come along for the battle, on the basis that the lord would look after them once he won (from the forfeited estates of those who picked the wrong side)
In the situation under discussion, a critical point was that many (perhaps even most) of the great vassal held lands under both both John and Phillip. In feudal theory, they could have (and should have) sent contingents to both . In practice Phillip broke the rules and upped the ante by demanding that vassals offer him /
liege/ homage. Liege homage was a much bigger deal than fealty. Liege homage, which seems natural to us nowadays, but was not so much so then, demands a personal loyalty to one lord, excluding all others. A bob each way was no longer an option.
The result was the lords who held under both overlords had to make a choice . Go the whole hog one way or the other. The choice, by and large, came down to an assessment of who was going to win. Arguably, if it had been Richard instead of John, quite a few might have decided differently.