I think there is a valid school of thought that (usually) interprets the number on a d20 roll as a scale that describes how well your character did something. For example, rolling an 18 on a Charisma (Persuasion) check is always viewed as better than, say, rolling a 12. Commonly, those who subscribe to this school don't tell players the DC for ability checks ahead of the roll and may not even have one in mind. The DM might be using feel and experience to determine success/failure in the moment based on the roll, with high being a success, low being a failure, and middling numbers going either way depending on how they feel the scene should go. This school gives the DM more narrative control. It also sets the expectation with players that a high roll is (almost) always a success and a low roll is (almost) always a failure.
Then there is another valid school of thought that (usually) interprets the number on a d20 relative to a static DC/AC or an opposed roll to simply determine success or failure. For example, rolling an 18 on a Cha(Persuasion) check is no better than rolling a 12 versus a DC 10. Both were good enough to succeed. This lends consistency to all d20 rolls (or "tests" as they might come to be called in 1D&D) - for example, a PC who rolled a 5 on a grapple check against an enemy who rolled a 3 did not grapple any worse that the PC who rolled a 17 vs the enemy's 3. Similarly, a PC who rolled a 15 on an attack roll did not hit any better than a PC who rolled a 10 against an AC 9 zombie. This school gives the players the knowledge of the outcome as soon as the dice settle.
Exception: For both schools of thought, a 20 on an attack roll is a crit and is a superior outcome to any other number. Well, as long as snake eyes aren't rolled for damage, I suppose.