So a big thing IMO is that CK3 is a character focused game and thus warfare should also be a character focused thing. What do I mean by this.
Say I am the duke of York and I have a rivalry with the Duke of Lancaster and his family. If he is a commander leading armies then I would want to be a commander, and ideally I would want my son if he is of age or one of my loyal vassals to be a commander. As one of the powerful vassals in England, I would demand this, and if the king did not give me my commands then I would be upset(lose opinion).
Since I am rivals with the duke of Lancaster, as we arrive to say the gates of Edinburgh in England's quest to conquer Scotland, I would want one of two things. First to have the right to lead the assault and get the glory and honor that comes with taking such a location. But if I can't be in charge I would sure as hell not want the duke of Lancaster and his allies in charge, I rather have the king or the duke of Kent in charge at least they aren't the perfidious Lancastrians.
Now instead say there is a civil war between the king of England's sons. at first I might be neutral chosing not to support either son. I don't really care who rules england as long as I get to rule the duchy of York. However the duke of Lancaster decides to support Prince Robert for the throne of England, well know I am pissed. I didn't care about this conflict before but now prince robert is suporting my hated rival, the duke of Lancaster? that just wont do. So I am very likely to make an offer to Prince William to support his claim to the throne of England so that the duke of Lancaster can't increase his power. Or if for some reason we are both fighting for prince Robert, then now Robert has to balance our hatred for each other. Knowing that if he favors one duke over the other that one of us might just bolt from the war and switch sides.
and of course being at war allows plenty of schemes and skullduggery to happen. If I was far more of an intrigue duke instead of a warlike duke, I might use the war to advance my schemes.
This is just a very basic sort of thing I outlined right here but the drama and character focus should be front and center.
Beyond that there are a few other things I think would be good.
Destructiveness
Destructiveness isn't here just in the chance of characters to die, but it is that. But it is about what warfare does to the landscape and towns. What do I mean here? Well, warfare is destructive to the people. Most medieval warfare was raiding with the battles and sieges sprinkled in. and that raiding was more often than not to the peasants then the lords themselves.
Then you have effectively mass slaughter. Things like the first crusade, the harrying of the north, the Mongol conquests and so on. Simulating this should be reflected. The bloodshed spilled in some of these conflicts is destructive. Lots of people die, people who could have been taxpayers, builders, farmers, smiths, soldiers, clerics, and so on.
Then you have the destruction of buildings. Right now buildings don't tend to be destroyed all that much outside of some events. You have looting and pillaging that destroys buildings. From things like Nalanda and the House of wisdom to smaller shrines and buildings.
A slightly related element is the spread of diseases. Diseases can be just as if not more so destructive than the actual armies that are doing the fighting. And diseases spread quickly and easily among armies, especially in this period. Long periods of prolonged warfare with lots of fighting will spread disease. This could actually lead to temporary truces or even ceasefires, after all if you are fighting a civil war and the king dies because he got sick on campaign. Well that changes things.