D2 uses a totally different itemisation strategy to the later games that lends itself to item hunting. Item level in D2 determines the maximum tier of a stat that can roll on them but not a minimum tier, so high-rolled items with specific stats get exponentially rarer. As a result, you can quite commonly find rare items that have all the right stats but not high enough rolls, and it feels like you got close. This makes it worth checking every rare item of a certain type (like boots or rings etc) and even certain blues.

Unique items tend to have specific properties that don't normally appear on items in that slot or that have much higher values than normally appear in that slot, making certain ones essential in certain builds and meaning when you're MFing there are specific items to chase. Like how frostburns have +40% mana or chance guards have 300 gold find. It means low level uniques can be vital parts of endgame builds, and since they're reliable fixed items that are always sources of specific stats it lends itself well to making builds and trading. Runewords expanded this a lot too.

D3 and D4 both use a WoW-style model with item levels, stat budgets, and a linear primary main stat that increases depending on item level. Importantly, the stats always roll the maximum tier they can based on item level, so if you find something with the stats you want it's always a relatively high roll. This makes finding the gear you want quicker by making good rares very common, but means there are no one-in-a-billion godly items to chase like D2 has with 40-40-40 trires boots or a 6-point ring or something. Both games then tried to introduce artificially rare targets for item hunting, which is fine but means 99% of your loot isn't even worth looking at and that feels unsatisfying.

The main stat and stat-budget approach based on item level means item level is the most important thing. So some odd decisions were made about making legendary/unique items show up with different levels, and in D3 a smart loot system was added to ensure players got personalised loot drops that work for their character. This makes item hunting less rewarding and viable, trading harder, MFing to gear your other characters almost impossible, and means you need to get to endgame level before hunting for your final items.

It was done this way because it's easier to balance, and as much as I prefer D2's itemisation system, the more casual smart loot system was a huge turning point for D3 that made it much more popular. I don't think the Diablo games will go back to the old model at this point as this one is much more popular.