If you're a crappy king the last thing you want is someone with a "legitimate claim" to the throne with an army to back them up instigating your court to depose you.
Followup question: do you know the story of how the Frankish Kingdom of Charlemagne the Great got split up into Lombards, France and HRE (basically modern Germany) ??
Kings juggled the conflicting tasks of choosing an inheritor, as well as preventing their sons from rising up and... let's just say "William the Bastard-ing" everyone.
Yes, on occasion a random bastard son will take over all of Normandy and Briton.
So I don't really know what your question is. That happened, constantly. Brothers killed each other, and sometimes their fathers, in a mad game of power for damn near a thousand years.
But clearly the kings of the time thought the risks were worth the reward. A hereditary title of king means that your followers know who is charge if you died in battle. The risk of your sons deposing you was accepted, because the risks of dying for other reasons (disease, famine, war...) was greater.
The way feudal societies worked, allegiance was like a (bottom up) tree. If your lord was at war with the king you followed the fool. ACOUP has a good number of posts on the workings of feudal societies if you have the time to read them.
Do you know why he was fighting in wars in far away lands? It certainly wasn't about land or power (at least, not for Richard)
It was a Crusade. The 3rd Crusade to be precise.
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Also the whole Reconquista thing in Spain, coinciding with the Spanish Inquisition and all that. Plenty of major religious events that interacted with politics and brought glory to commanders.
And then things like the 30 Years War, which had substantial amounts of religious influence of Protestants vs Catholics.
I'm not saying every war was about religion. But Catholics have PLENTY of holy warriors that are even venerated for their battle prowess.