In my opininion, the main problem regarding snowballing issue is the way AI lords behave on global map. They have nothing to do besides conquering. No feasts, no participating in tournaments, no solo patrolling their fiefs, no solo raiding, no nothing. A lot of this comes from deleting role of marshal and introducing create an army feature.
So my take on that. First of all, cost of gathering an army should be much more higher. Secondly, cost of containing lords in army should be a lot higher as well. Ideally army should disband after capturing 1 holding to give a breath for loser side. And at the same time higher cost of creating army gives time to lords to do their personal stuff (hiring new soldiers, patrolling, solo raiding etc) while they replenish influence to gather new army
You and I are actually on almost the exact same page here. It is way to easy for a lord to gather very large armies and it takes way too long for castles and towns to build up decent sized garrisons. Further, it is very easy to take a castle/town with 1:1 odds when you should need significantly more troops to take a castle/town as to defend it. I would almost go as far as saying the defenders should get like a 20% bonus to damage/defense just to make it harder to take a castle at least in light of how easy it is to pull 500-800 man armies together.
As for a player is shooting for an independent kingdom, they should allow the player to call his clan parties to make up an army even if he is not a vassal. That would give the player access to 5 parties of 60 + his own party of around 80 by Clan Tier 2. That is up to 380 troops which should allow him to take a weakened castle even if the defenders were toughened up.
As for influence, usually I am swimming with it pretty quickly. It think my level 17 character has like 3200 influence already and earn like 40 influence a day passively. I can basically summon any amount of available nobles at a whim and so can most to order lords in my Kingdom. Considering there really very few things to use influence on, they really need to make more influence sinks, adjust influence gain or just make things that cost influence most costly to make influence meaningful. One thing I wish they would add as an influence sink, is the ability to use influence to override decisions. For example, in an castle/town election, if they don't list you are a candidate, I would like to be able to spend 1000+ influence to make myself a candidate. Another would be to campaign to remove someone as a candidate and/or maybe promote someone you want relationship with as a candidate. Another sink might be to just being able to use influence to build relationships with other lords, i.e. pay 100 influence to X lord and gain 2 relations. There is so, so much that can be done with the Influence system.
I also agree about the lords not having anything to do. They need other activities to keep them out of wars. Maybe if bandit bands get too big, they start to actually raid villages or disrupt prosperity and trade to your Castles/Towns. In that case, you and the other lords would have to hunt down all the large bandit parties in the area. Of course those banquets, tournaments, solo raiding, etc are go things for the lords to be doing as well.
Finally I mostly agree about how armies should react. Once a castle/town is captured, they shouldn't immediately move on to another castle/town. Instead they should either sit in the city for like 5 days to defend it against it being retaken and/or patrol around the castle/city for the same period for the same reason, then they should disband to go take care of other things. I would also say that maybe any party that is reduced to 30% of their max party size should immediately disband to go build up their forces.