All that being said I still think that OSR game systems provide an incredibly resilient and robust method for running games and I don't think you would even really need dungeons for it to work. As far as I know, most OSR games function as either a series of dungeoncrawls or hexcrawls. This is one of the things that make them so thematically distinct from other "RPGs". While you can have intrigue and quests in OSR systems there are mechanical gaps in doing so. I think that one of the many reasons that they are so prevalent in other RPG systems which focus on a more narrative approach such as World of Darkness or Apocalypse World. Those game systems are designed with those objectives in mind and likely are able to mechanically incentivize their thematic elements so well because of that approach.
I don't think that a complete reworking of most OSR mechanics is necessary to achieve the ability to run games with a focus on quests or intrigue. Instead one can simply integrate those two elements to the central mechanic of a character gaining XP for treasure collected by converting quests and intrigue into treasure. One of my favorite videogames is Sunless Sea for a variety of reasons, chief of which is the way the game mechanically reinforces thematic elements central to gameplay. The game has a type of item which is referred to as "curiosities", unlike fuel and supplies these items represent meta objects such as secrets or knowledge. I've seen a few blogs bring up the idea of secrets and knowledge as treasure which may be sold to others, and I think that may be a way to seamlessly integrate intrigue into the general OSR XP gain mechanic. While it may be a minor paradigm shift for players to understand the value of hidden information which may be sold for wealth or leveraged over others in order to achieve something.
By giving players access to information brokers to whom they can sell secrets, a referee would create incentive for players to focus on the secrets and mysteries of a setting at relatively low effort. When creating NPC's or Dungeons a referee implicitly makes two sets of information. One known to the majority of people and one not known to many. People may know that there is a dungeon hidden in the swamp, but likely the majority do not know that there is valuable treasure inside or the tomb of an ancient warrior. Similarly, many may know that the chief of a tribe is named Mark but few may know that he is the bastard child of the emperor and would make a viable replacement to the current despot. I feel I've explained this concept adequately.
Generic Quests as Items Sold by Characters
The following table lists items which may be sold to information brokers or specialists and allow characters to gain XP as if they had gotten treasure from a dungeon.
|Terrain Map||25 silver per 5 hexes mapped out||Cartographer Guild|
|Monster's Head||10 silver per HD over 3||Wizard College/King|
|Outlaw's Head||5 silver per HD||Sheriff|
|Dungeon Map||5 silver per 10 rooms mapped out||Adventurers|
|Dungeon Histography||100 silver per level past 2||Historian|
|Secret||Varies (1d6 * 1d6 * 5?)||Information Brokers|
|Secret Spell Scrolls||20 per spell level||Wizards|
|State Secrets||Varies (1d8 * 1d8 * 10?)||Spies|
|Rare Ingredients||Varies (1d4 * 1d4 * 6?)||Pharmacists/Chefs|
|Tokens of Friendship||Varies (1d6 * 1d6 * 5?)||Kings/Jarls/Chiefs|
By allowing the characters to sell the following items you may incentivize different things. Terrain Maps would incentivize exploration in hex crawls. Monster or Outlaw heads would incentivize characters to confront foes. Dungeon Maps and Histographies would incentivize exploration of dungeons. Secret Spell Scrolls would incentivize robbing wizards. State secrets would incentivize espionage. Rare ingredients and Tokens of Friendship would incentivize exploration of possibly dangerous places by creating alternative "treasures". Rather than having a figure giving the characters a quest they would instead choose to attempt to get the quest objectives on their own. Ideally, this would provide characters internal motivation to do the quests for their own sake.