Our ideas of what the game should be like are informed by our preferred fantasy fiction.
The D&D rules are loose enough that you can "superimpose," for lack of a better word, a lot of different paradigms on top of them. I'm a much bigger fan of books like The Demon Princes* and The Name of the Wind than I am of, frex, The Wheel of Time or the earlier Dragonlance novels. Not every PC is going to be as big of a badass as Kirth Gersen, but he's somewhat illustrative of what I think a high-level PC should be like. And, essentially, that's what I like my D&D characters to be like. Even if it turns out that there's something I don't think they're likely to succeed at, they aren't at a loss.
If anybody happens to have a copy of the Chronicles of Amber around, bust that open and read the first few pages. You know what you'll notice? The protagonist wakes up in a hospital with amnesia. He's got a busted leg or arm, I forget which. Doesn't know his name or where he is. But he decides he doesn't like the cut of their gibberish, and he's stolen a gun, car, and money before you hit page ten. He doesn't fuck around. His GM isn't making him roll to see if he can use his Law and Intimidate skills in concert.
Anyway, it makes sense to use different approaches depending on what kind of campaign you want to run. If you want to run a game about peasant kids finding their place in the world and learning important lessons about friendship, absolutely, use 5E-style background packages and make them come up the hard way. If you are like me and you want PCs who have been around the block, then assume broad competence and only question it if what they want to do is very specialized.
Also, I really hate it if things happen like the following: The PCs are in a position to steal a ship and go become pirates. This sounds like a great idea to the GM. But then people start looking at their sheets, and nobody has the Command or Navigation or Seafaring skill, and it all shuts down. See, that's crap. Conan would have just stolen the ship and gone reaving.
What Attributes Mean
In later editions of D&D, it is very clear that attributes are supposed to have a great deal of correlation with what's going on in the game world. Like, it seems to be the intent that you could take the bell curve for Intelligence scores, correlate it with the real-world bell curve for IQ scores, and that they're intended to mean kind of the same thing. If you have a high Int, you have a high IQ. Similarly, if you look at your Strength score, you can probably figure it means something like what "being really strong" means in the real world. Someone with an 18 strength is assumed to be, like, an Olympic powerlifter or whatever. So on and so forth.
I'm not actually convinced that this was the intent in OD&D. Old Geezer, of course, is welcome to tell me that I've got this wrong.
There are actually two different kinds of ability score in OD&D, before you add in the expansions. I first started thinking about this when I read Lars Dangly's Platemail 27th Edition, where he said that PCs were much more like "toys" than the modern conception of "characters." Their actual game statistics were pretty vague and generic, and then you had to layer imaginative elements on top of that. It got me thinking.
For the sake of applying a label to them, let's call them Aptitude Attributes and Customization Attributes.
Aptitude Attributes are Strength, Intelligence, and Wisdom. Note that in OD&D, they're listed in that order. Those three come first. The only thing they do, mechanically, ismodify your XP gain if you are a member of that class. If you are a fighter, then your Strength modifies how fast you advance. Intelligence and Wisdom scores do essentially nothing. The reverse holds true if you play a Cleric or Magic User. If you play an MU, your strength is irrelevant. There's no damage or hit bonus or penalty. There's no modification to your encumbrance capacity. The Aptitude attributes are just that: they determine your attribute for various classes, and give you an incentive to try this class or that class at random.
The Customization Attributes are listed last. Dexterity, Constitution, Charisma. These three will actually give you mechanical modifiers regardless of class. They modify AC & Missile Weapons, HP, and Reaction Rolls/Henchmen/Morale, respectively. These can be important for any character.
How might this look in practice?
If you carried this idea farther than I ever did, you might do something like the following:
1. Roll your six attributes in order, 3d6.
2. Note any modifiers from your Customization Attributes. Once you've done this, erase the name and the attribute score. So, now you've got maybe a modifier to your AC or your reaction rolls, but you don't actually have anything called Dex, Con, or Cha on your sheet.
3. Pick a class. Write down the XP modifier for the relevant Aptitude Attribute.
4. Erase all those attributes and their values, too.
5. Yes, dead serious. Now you've got maybe an XP modifier, maybe a couple other mods, nothing else. If you rolled, for example, straight tens and elevens and wanted to play a Fighting Man, you might have a sheet that looked like this:
Berthold the Grim
Level 1 Lawful Fighting Man
Background: Berthold wandered out of the wastelands with a broadsword and a bad attitude. He has a cool scar on his face and wears his hair in dreadlocks.
Movement Rate: 9"
*Death Ray or Poison: 12
*Dragon Breath: 15
Equipment: (total encumbrance goes here, I don't feel like calculating it)
1 week rations & Waterskin
Belt Pouch with 10GP
Now, if I were to steal somewhat from Lars Dangly, and you were interested in what OG has to say about level being determinative of competency, here is what I would do for a skill system.
1. If a character wants to do an adventurer-ey thing, you either let them succeed, rule that it requires specialized skills they don't have, or make them roll.
2. If they have to roll, just have them roll D6 equal to their level. If they roll any 5s or 6s, they succeed. If it matters how well they succeed, count them up. 3 or more dice coming up successes is very good. If they're opposed by another character or creature with hit dice, make an opposed roll. You guys have done this kind of shit before in tons of other games, you can figure it out.
3. For things requiring special skills, just put a space on their sheet called "Skills." Whenever they do something for a substantial length of game time, they can gain that as a skill. If they are on a sailing vessel crossing the sea, they can say they help sail it. If they do that, just write down Sailing. If they buy a bunch of books on alchemy and read them over the winter, write down Alchemy. Don't make them jump through a bunch of hoops or anything, just keep track of what they've done. Keep things moving and fun.
4. All PCs are assumed to gain leadership ability in proportion to their level and other skills. If they know how to fight, they can lead fighting men. If they know academics, they can function as the dean of a university.
*This is some of the best sci-fi I've ever read. I recommend reading them at your earliest convenience, if you haven't read them before.