There are enough recent games that use ONLY abilities to resolve tasks: Knave* and TBH* come to mind. For example, rolling under Strength to break down a door or under Dexterity to pick a lock.
Moldvay Basic* DID have an optional rule ("there is always a chance") to indicate similar possibilities - using abilities regardless of level - but it also contained many thieve's skills that functioned using your level regardless of abilities.
I always felt a combination of both to be the most "realistic", sensible or even obvious solution. This (very popular) "stat + skill" combination would, in my view, be easy even for beginners, that would otherwise wonder "wait, how come my Strength 20 barbarian has no advantage when breaking down doors?".
However, lately I have been feeling that this is a bit redundant in terms of design. For example, a "Fighter 15" with a +15 attack bonus is easier an simpler than adding Strength to it.
This also makes monster easier to create and understand (i.e., a 15HD monster has the same bonus as a 15th level fighter - and the same HD, of course). Otherwise, you start wondering why only heroes get such bonuses.
Skills are relatively easy to "fix" - just disregard abilities, or choose between using ability OR skill. One advantage is that the smart rogue no longer beats the cleric at religion, for example.
Saving throws can be "unified" into a single category too - take Swords&Wizardy, for example - or give Fighters an edge at "fortitude" saves, etc.
But there are other effects of abilities that are more interesting, such as extra damage, extra HP, more spells or languages...
All these little +1s and +2s help to make each character unique, so no two 3rd level fighters are identical. One might be stronger, other more nimble - and they will PLAY different in combat, one avoiding more hits, the other delivering crushing blows, but not often.
As always, this is a case of preference. When rolling skills, for example, 3e and 4e rely heavily on level, while 5e is more balanced. Although 5e is my favorite of the three, I prefer level to have a larger influence, which is what I used in Dark Fantasy Basic.
But this "customizing" aspect of abilities is something I have no desire to get rid of.
Who knows, maybe I add "skills without abilities" as an optional rule...
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