I think we're too far down the development pathway now. But If I was to go back and do it again, I think there should be an overlap or merging between influence and relations. Relations should be the metric through which we are able to assemble armies. You should be able to earn relations through doing positive acts towards nobles or their lands - and that might include clearing their looters or donating them prisoners or troops or funding their city improvements or marriage or such like.
I am under the distinct impression that relationship scores affect the influence cost of summoning someone, but it's certainly not large enough to be noticeable and that factor should probably be boosted.
I haven't had as much of an immersion problem with influence in the past because:
1) Well I never actually farmed looters in the first place so that never broke my immersion.
2) I was using the Diplomacy mod which added a lot of options to use influence to - for instance - trigger an early end to a stupid war.
3) Diplomacy also added a "Take Over Faction" option which costs metric Ftons of influence and can only be successful if you've got higher influence than anyone else. So I ended up saving up a ton for that specifically.
4) I also used BL Tweaks to increase militia presence, which meant I always needed an army to take a town and usually the same for a castle too.
Quick thought: I think "influence" would be less incongruous if it was called "glory" instead.
Longer thought: It appears that - since EA launch - personalities and relations and persuasion mechanics have been a complete afterthought in development next to fixing engine issues, perfecting pointless combat grind, patching crash bugs and tweaking sheep textures.
I mean... hell... as far as I'm aware there has been zero work done on the PC's personality traits since launch other than fixing a bug which automatically reset your personality stats to zero the first time they were queried. And even that took until like November 2020, despite it being clearly an issue from the jump. It's still impossible to level Calculating, for instance, and breathtakingly difficult to level anything else.
This is really unfortunate. As I've said in the past, the lack of personality nuance in NPC's compared to even Warband means that - at a certain point - Bannerlord feels like an Excel spreadsheet with a fancy murder simulator GUI. It's just soulless, you know?