doctorbadwolf
Heretic of The Seventh Circle
I think that 5e D&D (i) doesn't have a clear system for extended PC vs PC social contest, and (ii) is ambigous over whether the outcome of a successful skill check is you did that well or you got what you wanted (roughtly task vs conflict resolution, without wanting to hang too much on tha parituclar terminology), and (iii) doesn't have a system for incorporating emotional/relationship components into a check.
I'm sure these things could be sorted out at a particular table, but I don't think they're there in the basic presentation of the rules. I think that makes it harder to get things going, or make some things obvious.
To elaborate on those thoughts a bit more: in D&D, if - during the course of the PC vs PC rivalry - one of the PCs suddenly escalates to violence, the whole arena of conflict is changed and there's no straightforward way to have the social conflict feed through into the new situation. Eg there's no obvious mechanic for the other PC to cow/shame the escalating PC into stepping down. I would generalise this point by saying - outside of some magical effects, and 4e skill challenges - D&D doesn't make it easy to establish finality in a scene simply via social interaction.
I also think that Prince Valiant "has everything named and described in a way that promotes a certain type of play" but I don't think that that is a separate thing bur rather is related to the features I've been describing above in this post.
All of this is before we get to the default XP-and-gp reward framework of D&D. Because, by default, social interactions tend not to yield either of these that also tends to make the seem secondary in play. 4e again is an exception, and you could easily drift 5e away from the default (though I don't think "milestone levelling" would necessarily help in this respect), but it is another feature of the game that differs from those systems that (as I've experienced them) tend to give the OP more fo what he's looking for out of the box.,
To finish, none of this is meant as a critique of D&D. It's meant as an attempt to reflect on how one might want to tweak/drift to get what one wants. If you - @doctorbadwolf - already have it then my thoughts are unncessary. But @Reynard did seem to be looking for some thoughts. (I hesitate to call it advice because I don't know 5e well enough. Maybe goal-oriented musings?)
I think perhaps my 4e experience makes the Tools 5e very quietly has more obvious to me than to many others.
So, one way to resolve shaming a rival into stepping down because their escalation is unbecoming, is simply to use skill checks in combat. It’s generally an action, but I’d allow it as part of taking the Dodge Action, if you wanted to take a defensive stance while shaming the offender into putting his damned sword away and acting like a knight.
Intimidation seems like a good fit, but I’d also allow persuasion or even Insight paired with Charisma.
As for letting relationship impact checks, there are many ways in the 5e rules to do that, they just aren’t explicitly called out (I really want a D&D setting book that is focused on romantic fantasy, that would have stuff like relationships, group morale, etc, with a more detailed and up front presentation).
*Advantage and Disadvantage obviously is the simplest way to model the impact of a relationship on a roll.
*Morale can be renamed to Fellowship, or Hope, or any number of other things, from the DMG
*Loyalty from the DMG or the crew quality score from Ghosts can represent a relationship score
A homebrew idea for such a thing, to add an emphasis on such things.
Create an additional ability score that is not generated at CharGen, but instead varies with contacts. Call it your Fellowship Score. You can have anywhere from a -3 to a +3 in Fellowship, and you have a score for each Relationship. You also have a Faith Score, which is the same thing, but for Factions.
When a check is related to a Relationship or a Faction (maybe call them a Loyalty?), you use your Fellowship or Faith score instead of another Ability Score and add a relevant proficiency mod.
as for rewards, I find it pretty easy to just...not use the standard reward structure.