There are plenty of things that could be done inside towns that would not require an enormous amount of effort and would not break other contingencies. For example:
Talk to a child and play hide and seek with them for 1 point in scouting, athletics or tactics and some XP that levels with your level so it is still worth doing later. It could also add some weight to the calculating trait. The child doesn't even need to find a place to hide, just have multiple places in the town where a 'hidden child' class can spawn. As an aside, town loyalty could gain a temporary boost.
Talk to a teenager and have them tell the player they need to deliver a letter to their relative. The target should ideally be one of the 'doors' in the town - since the player cannot activate certain doors but the label 'door' appears, adding a 'deliver letter' option for said doors will tie the mission up nicely. The mission should reward 1 point in steward or scouting and some XP that levels... etc It could also add some weight to the generous trait. As an aside, town prosperity could gain a temporary boost.
Talk to a townsfolk and gain a mission to clear an alley for 1 point in roguery or leadership and some XP that levels... etc It could also add some weight to the impulsive trait. As an aside, town security could gain a temporary boost on top of the default bonuses.
Spawn a lame horse somewhere on the map and have a mission from a townsfolk to return the injured horse for 1 point in riding, medicine or charm and some XP that levels etc. It could add some weight to the merciful trait and as an aside the town gains a temporary boost to loyalty or prosperity.
Beggars are another very easy class that could be used for side missions. Fist fights, fetch quests, help quests - the list goes on. I'd suggest focusing on the walls and towers of the towns for the beggar missions since they are severely under-used.
Beggar quests could either help or hinder a town - the player could have rogue, steward, medicine or charm options for quests depending on their current skills.
There are other things like this that could be done within the same town. Since they don't qualify as proper missions, they all fail if the player leaves town. I understand this would require a new module but since it's a self-contained system that fails if the player leaves town it would be very easy to implement. Getting the player to interact with the town in this way will add new levels of immersion, but players who want to skip all of that, can. There is no reason to have every side quest available in every town, but on average there should be one available in a town most of the time.
I could go on writing a lengthy list of somewhat plausible ideas but I won't; I will say that this example for implementation of immersion certainly isn't the only way to do it. One more thing:
It makes no sense to me that there are no options inside the keep. You cannot tell members to 'stay here', you cannot access the clan or kingdom pages and to be honest, if I'm in my keep I feel less powerful than when I am outside of it because I have no power there. A nice table with a current map of the world with coloured flags for towns and current battles would add so much for so little effort. It's something to do in the town that means something. Same goes for the 'manage town' or 'manage garrison' options - they should also exist inside the settlements.
It's unfortunate that TW seem to have catered more to the people who want to skip the world than those who want to enjoy it. I often feel far too rushed in the game and having reasons to go to towns will slow things down just a little. The same could be said for villages, which incidentally have no internal options for trading or hiring recruits (why?).
Last remark: Why can't I give beggars money, or shoes? It feels wrong not to have the option. Plus there you go - another side quest idea free of charge.