This board has a stand-alone and very active D&D forum. This, though, is the General forum. The OP is known to anyone who has read his previous posts as a fan of Savage Worlds. He didn't mention D&D in his OP except as a (late) example of a trend to have character-oriented bells & whistles as elements of PC build. He didn't ask for advice about how to play D&D.
And that's no real surprise. Discussing D&D is largely unhelpful to addressing the issues raised in the OP.
And ignoring D&D while addressing those issues is largely unhelpful to the vast majority of readers.
If I want to run a game that will give me the feel of Ben Hur, it's probably a limitation if the system doesn't have any way to resolve chariot races, or interactions with prophets and holy men.
Resolving chariot races need only draw on whatever that system might use for movement rules or chase/pursuit rules, modified by the GM to suit the situation.
"Resolving" interactions with prophets or holy types rolls right into the point I'm trying to make: those interactions can occur and be roleplayed through but IMO without external pressure (which would almost invariably drift quickly into combat rules and-or GM fiat) they cannot ever be resolved in finality.
You meet the prophet, you hear what the prophet has to say, and the meeting for whatever reason ends. Even if that meeting can never be repeated (the prophet dies, or is no longer accessible, or whatever) it's still entirely up to each individual character* what to make of that meeting, and whether or not to act on anything the prophet said (or didn't say).
* - perhaps in discussion with others in the party, but this isn't mandatory.
If the OP wants to run a character-driven game, it's a limitation if the system doesn't have the mechanics needed for the player to try (and perhaps fail) to put his/her PC's mark on the gameworld. If that domain of finality of resolution is confined to combat then that is going to be a pretty big bar to character-driven play.
Here I suppose it hinges on what you mean by "mark on the gameworld". If you mean something like a PC trying to rise to become Empress of the realm or a party trying to overthrow a barony* then I think we're talking about similar sorts of things.
* - by the party's own choice; though this could just as easily be a GM-guided plot.
If it's always open to the GM to re-open the situation, re-enliven the stakes, treat nothing as resolved, then there can't be character-driven play of the sort referred to in the OP - ie the mechanical resolution of action declarations generating a dramatic arc. There can only be GM decision-making about what story to establish.
This is just it, though: in any roleplay situation that doesn't have an artificially-forced closure it's also always open to me-as-PC/player to re-open the situation, re-enliven the stakes, and-or treat nothing as resolved as long as a) those involved the first time are still around i.e. not dead or far away, and b) those involved are still in control of their own thoughts and-or opinions i.e. not charmed or otherwise mechanically restricted. If I don't agree with what the Duke had to say the first time before he shut me down, I can always try to talk to him again - maybe he's changed his mind or had second thoughts. By the same token, if the Duke doesn't like the answer we-as-party give him when he tries to send us on a mission, he can always try to ask again. There's no hard closure on these sorts of things, and yes, sometimes it can result in things going in circles - just like real life.
This no-forced-closure idea holds even more water if the situation mostly involves one or more other PCs rather than any particular elements of the setting and-or its NPCs.
Example: I-as-PC might be trying to talk the party into chipping in funds toward a castle for use as a home base. Should a game mechanic be allowed to determine whether I succeed or fail? Of course not! It's up to me to roleplay the request in character in such a way as to get the response I want, and up to the other players to respond as they would in character to the request. And if they say 'no', a game mechanic should never prevent me from trying again later or - if I'm less wise - from continuing to badger them about it till hell's half frozen over. Again, though, I stress that the GM has to allow however much time it takes for this discussion to play out; and not get impatient.
Or, in a different vein take two PCs (or groups of) who have for whatever reason established a non-deadly rivalry of one-upsmanship within the party that drives most of their in-character decision-making. Regardless what the GM might put in front of them, any decision those characters make is likely to be filtered through how it might affect that rivalry; perhaps even getting to the point where the GM's hooks and stories are ignored in favour of acting in furtherment of one side of the rivalry: "You met the Duke, did ya? Gave us a mission [GM hook]? Well it's just going to have to wait, because I've got a date with Prince Vonwe next week and anything he wants us to do will of course take priority." (
I saunter off, and then frantically start pulling every string I can to get this date 'cause I was making that crap up and if I don't deliver I'm screwed! And even if I get the date, if Vonwe doesn't have a mission for us I'll have to dream one up because there's no way in hell that popinjay Fighter's setting our agenda if I can help it!)
There's no way in the world a social game mechanic should ever be able to step in and say "Sorry, rivalry's over in finality, here's who won".