So then why did damn near every European royal family make a habit of giving every prince and bastard son, including ones not in line for the throne, armies to go play around with?

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.

Good question.

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.

Because the parent's central thesis is wrong. Kings didn't war to spread religion because most of them weren't religious. They warred over land and power.

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.