What do those games you mentioned do to support such play?
They don't treat
treasure/income as a component of PC build. They have social resolution mechanics. They allow time to pass (though in different ways - in RM, because healing, studying rituals, etc takes time; in Prince Valiant because all passage of time and recovery and so on is purely GM fiat, and so it is easy to narrate "seasons pass").
Another feature that I think is important is that they downplay the idea of "the adventure" and "the quest giver". Without wanting to be too pejorative about D&D - which is an unhelpful turn that this thread is taking - I think it's fair to say that they depend a bit less than D&D often does on "unreal" contrivances to make the action work. The sorts of contrivances I'm referring to are the dungeon, wacky traps, sequences of monsters, etc that tend to characterise a lot of D&D play. I think this sort of stuff crowds out "realistic" fiction.
In 5e play the idea of 6 o 8 encounters per "adventuring day" is one part of cross-class balance. I think that can be one source of pressure towards the contrivances I've just tried to describe. So that's one are you might look at - eg will changing recovery periods allow play to take on a more "realistic" rhythm. (I don't know the answer, but am just trying to think some things through.)
In my Prince Valiant game, when two PCs were rivals for the hand of the one maiden, we were able to resolve their courtship in various ways because the game has a uniform resolution system for all sorts of conflict - I can't remember all the details now, but I do remember at one point the two PCs were deciding who would be the one to be sponsored by Violette at the joust (or something like that) and resolving it via opposed Fellowship checks. This is an example of what I mean when I say that the "realistic" action isn't something that sits outside of, or alongside, the real action of the game but rather is itself part of the action.
I don't have a clear idea of how one would drift 5e D&D in this sort of direciton - I don't know the system well enough. I haven't tried this sort of drift in 4e D&D - our 4e game (with many of the same players as these other games I've described) didn't have a lot of this sort of grounded realism in it: it was more about charismatic leadership and cosmological conflicts than friends and family and interpersonal relationships.
But I would look at how one frames the action, how one treats treasure and other rewards, what one puts at stake in the fiction, how one handles pacing - these are some of the things that seem relevant to me.