In modern D&D, your class gives you most of your abilities as you level up: higher level spellcasters learn more magic, higher level fighters learn more maneuvers, etc.
However, earlier D&D had less "class abilities" in it, and those that it did have (such as wizard spells) were not guaranteed: you may fail to learn a spell, or fail to gain an item. It can even be argued that, in early D&D, a fighter's class abilities were basically its tremendous list of things it was proficient in: this enabled it to use more magic items than any other class.
The game has gotten away from this model, toward a model where your class gives you multiple abilities as you gain levels. However, it may be possible to return to an advancement system that is more treasure-based.
Every Ability Is An Item
The first big change is to eradicate the idea of gaining new class abilities (maneuvers, spells, or otherwise) as your character gains levels. You will not. Gaining levels might give you vertical advancement (your +1's become +2's), but it won't give you horizontal advancement (you won't learn any new abilities).
Instead, turn your usual class abilities into items. For instance, a fighter will not learn a maneuver that knocks an enemy prone. Rather, they will have a magic item that enables them to knock an enemy prone. Likewise, a wizard will not learn fireball. Rather, they will have a magic item that enables them to detonate a ball of fire.
These items must be part of the usual treasure awarded to a character by undertaking adventures. Thus, player abilities become more reliant upon DM's placing them. If you don't want your wizard to ever have Teleport, it's easy just to stop them from ever coming across an item that can teleport people. At the same time, though, if a DM doesn't place any items in the game that have abilities, the characters will never learn new abilities: a wizard may never learn magic missile unless the DM opts to include it.
Item Slots & Abilities
If objects are to be the main location of a characters' abilities, knowing how many objects you can take on an adventure with you is key. This (re-) introduces the idea of item slots: you can equip only one thing at a time on your head, and whatever ability that helmet or crown or ioun stone may have is mutually exclusive with other head-based items.
What specific slots exist are open to much DM leeway, since this is "horizontal" advancement. A DM who only wants players to have a handful of abilities might have a single "ability" slot (regardless of how much equipment you have). A DM who wants to concentrate on combat options might have a "Weapon" slot and an "Armor" slot (for attack and defense, respectively).
No, You Don't Get To Choose
This model is very "in-play-centric," and thus skewers peoples' abilities to have character "builds." No longer can you revolve your wizard around a certain class of magic, or specialize your fighter in one specific weapon. Rather, what you gain as you adventure becomes your abilities, and it probably won't line up with a tight theme or mechanical trick.
Whether or not this is a positive or a negative thing can vary drastically with the group. Building a character is half the fun for some players, while others disdain the kind of min-maxing and math-tweaking that can come from people tinkering with things away from the table. The player doesn't have many options themselves outside of play: they only get to choose from a menu of things the DM introduces to them specifically in play. For players who like to build their characters away from the table based on their own idea of how their character should be, this option is not great. For those who enjoy building their character primarily through play, this option is quite positive.
Mastering an Ability
In this model, a characters' current capabilities are very tied to the equipment they're using: a fighter with a Scourge of Wounding might get to make an attack that bleeds for extra damage, but give that fighter a Hammer of Stunning instead, and they can't do the bleed attack any more.
This can become an unwanted characteristic. A character in this mode who meets a rust monster might not just lose protection and attack capability, they might lose an entire suite of abilities, permenantly.
As an option, you might want to allow PC's to "master" one ability of an item they have when they gain a level. For instance, a fighter with a flaming sword might master the ability to forge flame into any weapon they have. A wizard might learn Sleep so well they can cast it once per day without access to the Sandman's Bag that enables them to cast it otherwise.
One way to export a lot of the mental effort involved in assigning items to the party is to randomize a treasure award that you roll when the party completes some task and is offered a reward, or when they plumb the depths of some ruins. You could even assign an "XP value" to these rewards: a budget of XP to spend on monsters and threats that, when defeated, grant that randomized award. Regardless of your method, this allows you to not bother selecting a specific treasure, and instead to cull it from a list of treasure.
Who uses what?
It is possible when using this system to disregard a more formal class system. Rather than dictating what abilities your character receives and can use based on the class, actual play helps shape your character. Perhaps they use swords and wear robes simply because these are the abilities that are attractive to them.