I don't even remember how the supply system works, I think just the same boring ass "stay below the limit or suffer attrition" thing, but yeah, any Medieval army should be like this thing that eats up finite supplies in a province and in the process (especially at low levels of supplies) wrecks the economy of the province. I can easily imagine Chevauchee as a setting you can flip the army too, like flipping it to Looting in CK2, which makes the army destroy way more Supplies than it actually needs in order to quickly rack up that damage on deep penetration raids.
This also is, much like seas freezing in winter, something that maybe AI is too stupid to handle (Redditors DEFINITELY too stupid to handle), but didn't armies even as late as the 1800s avoid fighting in Winter for the most part? And even oftentimes disband and go home for the harvest? "Campaign season" should really be a thing. Even Total War makes an effort to disincentivize mindlessly roaming around enemy territory in the snow.
Sieges especially manage to be surprisingly lame given that it's the one era when they're the most iconic. In my fantasy CK3 (the good one, not the one that actually exists), sieges would have like a Wall integrity stat and if it's above a threshold that acts as a multiplier on army strength, siege engines take a while to build (construction starts automatically, dedicated engineers in the army speed it up and make better engines) but reduce Walls. Settlement also has Supplies that drain down. Surrender happens as a chance when Supplies are low and Walls take a long time to recover. Certain special statuses - breach in wall, cut off water supply, and cut off supplies in general - would effect these (if a defender can keep a supply channel open they can greatly prolong the siege).
When you combine that with supplies and plague for the besieging army, suddenly you get a situation (as in real life) where the sieger is on just as much of a time table as the defender, because when you're sitting in place with no logistical networks you could very well run out of food before they, with their storehouses, do.
Big thing, make siege much more interesting: have the ability to sort of gamble on it with threats. I've heard that it was customary in the Medieval period to threaten to obliterate a city if it didn't surrender immediately, often brought up by Crusader apologists for the Sack of Jerusalem. So, you roll up on a city, you get to set a promise of max looting if they do surrender and min looting if they don't. Big thing, if you make a severe threat they will be more likely to surrender, but if not then they'll hold out LONGER when they do fight (because it's so catastrophic to lose).
EU4 sieges currently feel much more intense and "epic" than CK2 sieges, probably because they are less predictable and frustrate through getting in your way.