I'm familiar with Madden's more political stuff. I also read his book on the Fourth Crusade.
One thing mentions a lot is that our understanding of Crusades is heavily influenced by 19th century colonialism. "Our understanding" being both the modern West and modern Islamic understanding.
It's also completely and totally wrong.
Just because a bunch of Christians and bunch of Muslims fought, does not mean it's a crusade. And just as there were no Crusades in the 19th century (with one teeny-tiny exception) there were no Crusades in the 9th century.
What's most relevant this conversation is that ChatGPT would be opening itself to lots of criticism if it started talking about 9th Century Crusades.
There are simply too many reputable documents saying "the first crusade began in ..." or "the concept of crusading evolved in Spain ..."
I'm reaching into my memory from college, but I recall crusading was mostly a Norman-Franco led thing (plenty of exceptions, of course).
Papal foreign policy was based around one very simple principal: avoid all concentrations of power.
Crusading was useful when it supported that principal, and harmful when it degraded it.
So the ideal papal crusade was one that was poorly managed, unlikely to succeed, but messed up the established political order just enough that all the kingdoms were weakened.
Which is exactly what the crusades looked like.