Well, I'm not going to dig through old modules to prove my point. And, who knows. Maybe I'm remembering incorrectly. It was a long time ago. But, in the early AD&D years, I do remember interesting uses of dice for things--obstacles in adventures. A roll to climb up a mountain (not using the Thieving skill, but for the whole party), or a roll to avoid a trap...or to make it across a rickety old bridge. Or, just a straight roll for things like, "roll 1d4, where a roll of 1 indicates that the character smells the....".
That last is extensively used in early D&D mechanics, at least pre-Moldvay. It's the same as is used for encounters.
I had some time, and some materials on the harddrive... and couldn't recall any stat-or-less rolls. Early D&D is a LOT less defined than, say, early Traveller.
Let's compare the 1977/78 editions of several games... and the 1974 D&D.
D&D-74: lots of 1d6 for 1-N, where N is a chance set by rules or the DM. Some rolls of 1-N on other dice. Saving throws in 5 categories, unmodified by stats, 1d20 roll high against a TN set by class and level. Combat rolls in the "alternative combat system" 1d20+statmod against TN set by table lookup of Class/level vs target Hit Dice or target Class/level; the "chainmail" combat system, officially "The" combat system, but far lesser used 2d6 roll high, modified by attribute. Attribute modifiers only +1 or -1; on 1dX vs 1-N, and only 4 attributes had explicit use that way: Dex, Con, Cha- Dex for missile to-hits, con for HP, and Cha for loyalty. No advice on devising other throws
D&D 77 (OE + Sups 1 & 2 & Dragon issues 1-10): percentile skills added (thieves, rangers, bards, assassins). Several of the rolls changed from 1d6 for 1-N to other types of die, almost always 1d8, 1d10, or 1d12. Drop the chainmail option for combat. Open Doors now a stat dependent range. Dex, Con, Int have percentile rolls derived from table lookups by attribute. Str now has explicit open doors range and adds to to-hits in melee, dex has wider to-hit mods. Thief, Monk, and Assassin have percentile roll skills. Advice on deriving percentiles from attribute by multiplication for difficulty in Dragon. Several additional classes with percentile abilities.
AD&D: Every attribute has some percentage roll, and some modifiers to other rolls. Reaction rolls modifiers added to Cha. Bards & rangers added, also with percentile "skills".
Traveller 77: all rolls on d6's. Mostly 2d6+mods for TN+, combat always TN 8+. Modifier value of a skill level not consistent; some are if skill n+ then+Y, others are +Level or +2x Level. A couple adventures use 2d6 for attribute or less, one uses 3d6.
RuneQuest: all skills percentile, attribute tests percentile vs 5x attribute. Difficulties as modifers to score.
Starships and Spacemen (78): all ability rolls 1d20 for Attribute or less, attribute modified for difficulty. Rolls to hit only for ranged weapons, and are modified attribute throws, doing rolled damage to St. Starship to hit numbers fixed by range, not ability. Close combat is 1d6+Strength, and compared to opposing roll of same, then looked up in what should have been a table (but is a paragraph). Ship combat to hit is determined by range, not abilities, but is modified for class.
Tunnels and Trolls (original is 74, but the crica 76-77 4th ed version is what I'm basing upon): two mechanics: weapon's dice+adds + mods from attributes (Str, Dex, Luck) for damage total; difference between sides damage to losing side, allocated among them. Saving rolls officially always Luck, and 2d6 doubles open end +Attribute; option to use saves on other attributes besides Luck. In the '79 5th ed (which remained the ed in print until 2003!), saves on other attributes standard, and functionally a task system. Attribute mods for combat are Att-12 for bonuses, or attribute-9 for penalties, thus 9-12 is no modifier.
Space Patrol: Attribute mods same as T&T for a number of rolls and situations (Movement, for example, is modified by Strength). To hit with ranged weapons is a table based upon dex and range, using 1d6+ mods, roll low; Table is memorizable, tho, as short range is dex/3 or less, rounded up, +1 at point blank, -1 at medium and -2 at long. Melee Combat (called "Contact Combat") is (weapon)d6's (unarmed is 1d6) + attribute modifers from Str & Dex vs 1d6 plus attribute modifers from target's strength and dex, difference being damage.
Boot Hill (1975): percentile abilities... except strength which gives an 8-20 based upon the percentile as well... but they aren't rolled directly. All the combat process is base chance modified by a table lookup. The reason for the percentile attributes is experience... Melee is a 2d6 throw; shooting is a percentile throw. Gambling is a special ability, and is rated with a percentage chance to be caught cheating if cheating; this chance is random.
We can see that there are a lot of different methods in use, but already several systems have prototypes for later task systems.