As an example. Int always adds more to a Wizard than it does to a Fighter. Strength theoretically adds the same to both fighter and wizard, but in actual play it's benefits help the fighter more (fighters spend more turns attacking, they have heavier gear to carry, etc).
Thus, IMO if the abilities are balanced independently of classes then when we layer in classes they will cease to be balanced because classes use abilities differently.
*But couldn't we give each class different abilities tied into each of the various stats to solve this? Sure... but in this state the whole point of having common themed attributes is lost because they now impact each class differently - and so we might as well just tie those attribute based abilities to the class itself instead of jumping through all the mathematical conversions to get attributes to ultimately place us in the same spot.
When balancing the abilities, it is always the conceptual structure, the 'deep four': Strength, Dexterity, Charisma, Intelligence. More precisely, it is their aspect of the 'deep four' Saves: Fortitude, Reflex, Will, Perception.
The reason why six or eight abilities are possible is because the Attack bonus is so important, by itself it is worth an investment in an ability.
The abilities include traits for 'A' (Attack and Spell DC), 'S' (Save and AC), and 'U' (Utilities being various other traits such as Hit Points or Skills that are useful in combat).
When there are four abilities, every ability additionally serves as the Attack stat:
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When there are six abilities in balance, the Attack stats separate out as their own ability.
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When eight abilities, each of the deep four abilities has its own corresponding Attack ability.
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Thus the quest to balance the abilities is strictly about balancing Saves and Utilities. The deep four Saves are: Fortitude, Reflex, Will, Perception. Because there might be more spells or monsters that refer to one of these Saves, some might be more frequently useful than others. Generally, Reflex is the best save, followed closely by Fortitude. But it is easy to make the mental saves more appealing by adding combat Utilities.
With regard to classes, it is the Attack stat that is so important. Fighter relies on either Strength or Dexterity. Wizard relies only on Intelligence. And so on.
However, every class depends on the Saves and Utilities. Even a Fighter will waste a feat to shore up the Will Save proficiency. Especially the Wizard depends vitally on Reflex AC and Constitution Hit Points, and is actually a Multi-Ability-Dependent class.
In sum, balancing the abilities (whether they are four, six, or eight) is all about balancing the Utilities-Saves for each of the 'deep four' abilities.