One of the commonly touted tenets of OSR play is "diegetic progression"; most character progression is in the world, not on the character sheet. This is why you see a lot weird magic items in OSR resources, stuff that gives particular abilities that encourage creative uses rather than just a + to stats. This type of progression also includes relationships with NPCs and factions that players can call on for assistance. Players don't need to be able to defeat the Ogre in combat if the local chieftain owes them a big favour for rescuing his daughter and will send a squad of his best soldiers to fight it for you/chase it off.

Not having skills on the character sheet is one of the core ideas of OSR play, the idea that players should be coming up with creative solutions in the game and not just relying on the pass or fail of a dice roll to solve problems. Hand in hand with this is, as the above commenter mentions, "rulings over rules" which emphasizes the GM making decisions about how player actions play out in the world rather than looking for mechanics in a rulebook. This encourages stuff like creative tactics in combat, e.g. a player tips over a bookshelf onto the group of goblins; the GM decides the goblins next to it have a 50% chance of dodging out of the way or getting knocked down, or players have advantage against them on their attacks next round as the goblins dive out of the way, etc. There's no rules for this, so the sky is the limit for players to try out cool ideas. Players stop looking at their character sheets and rulebooks when presented with a problem in the game, they engage harder with the game, usually asking questions about details of the situation to see if there is anything they can use to their advantage. For groups that embrace this style of play it is much more immersive than playing a game where your options are dictated by game mechanics rather than the game world.

The most important idea in this, imo, is that a lot of stuff shouldn't even be rolled for; if its reasonable that a player could do it, then it just succeeds. If the players come up with a good idea, just have it work for them unless they are under stress (e.g. hurrying to pick a lock while a boulder rolls down the hallway at them) or there is some adversarial element where an opponent's skill could counteract the player's, e.g. seeing through a player's disguise, avoiding 2 players trying to wrap them in a rope, etc. This really incentivizes the players to think creatively, when their good ideas are rewarded without being at the fickle whims of the dice.

Dungeon delving is probably the default mode of play, and best supported by the rules, but that does not mean it has to be at the center of every session. characters still have a charisma score for social checks, they still have backgrounds that can give them advantages in a given situation, if the dm grants them, it's just more free form and less guided.

for example one of my players rolled up a thief with pretty high charisma with the "noble" background, we decided that he belonged to a lower house before adventure called and he fell into disgrace, nothing major but with enough renown that his sigil ring would be recognized in parts of the world which gave them an in with an opportunistic mayor hoping for a favour. roles where mostly used if i was not sure how the mayor would react to offers of vague promises in exchange for information or making sure the city guards would not be patrolling the harbour during the night so they had a chance to inspect the shipment of a merchant they believed to be a supporter of an evil cult.

stuff like that does not need much support from the rules, just something to resolve questions the dm does not know the answere to and people being able to improvise a bit. neither of us knew much about that low noble house, i don't even think the sigil ring was my idea, but they are now a thing. they are magical and hurt everyone trying to wear them who does not belong to the house and my players house apparently controls some fruitful tracts of lands and is known for a very sturdy horse breed.