They could haven given each class one stat that was useless to them, like Intelligence is not good for Barbs but Str/Dex/Vit are good.
Or a better option would be to pick 4 aspects of each characters performance and tie that to a stat. These aspects would have to be functionally different. Not all adding to damage or all adding to survival. But each one would enhance each class, just differently.
Str: increases melee and some physical skills,
Dex: increases movement and physical projectiles,
Int: increases some skills, magical projectiles and some resource pools,
Vit: increases Health, Resists and some resource pools.
The player then chooses what becomes their primary stat by deciding which aspects they want to emphasize. This could result in Int barbs and strength sorcerers, but so long as there are skills and builds that are enhanced by these choices, it simply creates many new ways to play a character.
I think pure stats should have been stripped off of gear and only have been found on gems. Gear should contain all other non stat modifiers. This frees gear up considerably. Now you would look for modifiers that support your skills and build, then socket it to make it reflect what stats you desire.
It's important to see modifiers that multiple skills and classes would want to utilize. This could be done by modifiers that apply to the types of actions a player can make. Shoot a projectile, move, block, take damage, heal, spend a resource.
It is important to create similar modifiers that heavily compliment one another. Like a modifier that increases fire damage, one that increases the chance fire damage burns enemies, one that lowers enemy fire resist and one that increases duration or intensity of burn effects. This way, getting an item with one of those modifiers is useful, getting an item roll with all of them is godly.
Some Legendary weapons should have unique modifiers that open up entirely new playstyles and skill options not found anywhere else. Flip flopping health and resources values, projectiles going through enemies, granting unique skills only found on that weapon ect.
Thing is, Itemization IS the game in Diablo after the first playthrough. The higher difficulty levels only exist to extend the item acquisition process. For them to screw it up this hard is just mind bottling. I mean what they hell were they doing if not making a robust loot system. Working on the story?