Strictly speaking it doesn’t need one. What it does need is some unifying ideal (in the positive sense, not the current negative one) to inspire it. That’s most easily and effectively delivered in the form of a man who can inspire and work against fundamentalism. I don’t think it was a total coincidence that the end of Caliphate coincides generally with the rise of fundamentalism. Without a unifying ideal and in the face of increasing despotism, people turned to that which promised them an answer - a return to a “golden age” or the ability to blame all their troubles on foreigners.
No. First off, the religious heirarchy was not particularly interested in the petty conflicts of chieftains and warlords, who in turn were not initiatally Christian (and many Christians were Arian, not Catholic). The Church was generally uninterested in wars because they killed good Christians with no benefit - and dangerous in the face of Pagan and eventually Islamic aggression. Though it was better to have open wars than simmering permanent skirmishes.
Second, the “Dark Ages” saw many innovations in shipbuilding, bridgebuilding, architecture, and agriculture, as well as new themes in literature and forms of music. It was trade and formal education which declined, replacing a conservative, imperial Roman civilization with a younger, more active and flexible Germanic one. The cultures mixed in different degrees in different areas, of course, so that France and Spain were less germanic than, well, Germany, and also England.
Third, warfare did not advance terribly much during the Dark Ages. Advances in metallurgy made certain weapon forms more popular, which in turn influenced tactics, but a bigger change was simply the new germanic social order, where powerful lords claimed lands and eventually titles in accordance with their power. As under the original (and long-gone) early Roman and Greek models, only relatively wealthy individuals could afford quality arms and armor, not to mention horses or the time to pratice with their gear. So they became a warrior elite.