I'm not going to say it was the pinnacle of immersion or anything, but in most ways other than visuals, it was/is more immersive than Bannerlord.
Already have played two full campaigns of unmodded warband, plus multiple partial ones doing stuff other than map-painting. It is boring to conquer the whole map piece by piece once you've gotten to full snowball (Bannerlord has exactly the same problem) and of course it's not the most immersive game ever, but that aside, the process of getting there is more immersive than Bannerlord in lots of different ways, most of which are in the link in my signature which you've already seen.
I think the point is that despite Warband not being perfect, Bannerlord is a step backwards from it in a lot of ways, instead of an advancement.
Also lord dialogue had more of the lord's personality in it, and claimants who tell you stories of your kingdom and ask you to join them in a civil war, being ambushed by bandits/assassins who were hired by lords who hated you, fighting drunkards in taverns, companions telling you stories about themselves and Calradia that present points of view from almost every group in the game, commanding your allies to attack/defend/retreat to a specific location without your presence, roaming deserters and manhunters who made the map feel more alive, booksellers who sold books you could read to gain skills, being able to ask lords why they were doing something and get an answer, the ability to pitch a camp while waiting, having to actually fight your way out when failing to sneak into a town, vassals coming to your kingdom's court to offer their services, political quests to actually instigate a war yourself by robbing caravans or denounce other lords, resolving disputes among your lords, court gossiping to learn about a lady's personality and coming back to visit her during courtship, very basic casus belli for wars like "Citizens of the Kingdom of Nords say they are suffering mistreatment, and petition King Harlaus for protection. This will make it easier for him to go to war, if he wants it..." fighting duels with other lords when courting a lady or denouncing them, asking peasants to light a fire in a village for you to create a distraction while you did a prison breakout, and battling in the street and keep during a siege- among other things.
The multiple broken, incomplete, or unimplemented mechanics in Bannerlord don't help immersion either. There's a lot about Bannerlord that's good, but it's still missing a lot of meat on the bones which Warband had.
It's slightly too common in Warband, but it's much more realistic than throwing a random group of criminals, nobles, and scholars together for long periods of time and them all living in perfect harmony.
I genuinely enjoy it sometimes. Like in Viking Conquest, Donnchadh straight up tells you that Welshmen (iirc?) ruined his life and he won't put up with having one in your party. I thought "hey, I bet that's going to happen eventually." Then I did hire a Welsh guy and of course he got mad, and I enjoyed taking Donnchadh's side and telling the Welsh guy to shut up, since I liked Donnchadh.
It's another choice for the player to make- which companion do I value more? - that provides a diversion from the constant grind of looters. It reminds you what your companions' personalities are, and makes them feel more like people joining your journey instead of caravan-leading robots.
Getting higher-tier troops was much easier. When it comes to skill grinding though, I do like Bannerlord's skill system more, and find it more immersive than Warband.
Yeah. I'm still waiting to be able to run a "highly profitable" criminal enterprise in alleyways and base my gang there like they said we were going to be able to.