Well... let's see what Mr. Moldvay said to do... as, when I started, that's where we started from...
Lots of references like this:
Find concealed doors: search 1 turn, 1 on 1d6 chance
For other stuff, Moldvay gives several options, but the first is figure out the percentage chance.
Then he goes on to suggest basing it off of an ability score; roll 1d20 for attribute or less. The example is Rope Climbing by non-thieves.
He also suggests a difficulty mod of from +4 to stat through -4 to stat.
And the resultant play mode
If there was an explicit rule, we'd look for it. Fortunately, Moldvay was short.
If there was a specific example in the rules, that became a second tier of rules for us.
So, we had two "tiers" of climbing... 1d20 for rope climbs, and 1d100 for free climbs... both on dex for non-thieves, and on Climb Sheer Surfaces for thieves.
We occasionally based that 1d20 check on level instead. Other items we used 1d10 vs level, or even 1d6, and usually, a ±4 difficulty, and the stat modifier from a relevant stat if we could justify it to the DM.
Of course, we also used lots of saving throws... The hot princess just blew you a kiss, save vs paralysis.
The general rule was, if it was in doubt, we rolled for it. Movement seldom was in doubt.
Looking at the earlier editions
I started a bit later than these, but several friends who started when I did started with Holmes Basic or even with later printings of Gygax & Arneson.
Lots of specific "25% chance" type rules, and many more "on 1 roll of 1" (presumed to be on 1d6) references, but no general advice to make attribute checks. Attribute checks, however, seem to have been normative. Saving throws work just like later editions.
Holmes (p. 40) adds a precursor to Moldvay's generic advice:
Improvise. Agree on a probability that an event will occur and convert it into a die roll — roll the number and see what happens! The game is intended to be fun and the rules modified if the players desire. Do not hesitate to invent, create and experiment with new ideas. Imagination is the key to a good game. Enjoy!
It seems that Gygax and Arneson D&D left out some elements that (I've been assured) were so common sense that they didn't realize others wouldn't.
Other Early Games
Tunnels and Trolls: The mechanics of the 1975 T&T are, in fact, almost exactly the same as the mechanics of the 1979 edition 5.0... and the 2005 edition 5.5 simply adds an options section in the back, and is the standard ruleset still sold by Flying Buffalo, new in box. 7.5 is an alternative ruleset, closer to what Ken runs now; the core mechanics aside from character generation and advancement are pretty much the same as 5.0. A few small changes.
Best bet for the feel? Head over to DriveThroughRPG.com, and download the "Tunnels & Trolls Free Rulebook"... it's the T&T equivalent of Moldvay... the stable 1979 rules (T&T 5th ed) abridged into an introductory ruleset. It's more than "just enough"... it explicitly allows playing warriors and wizards.
T&T was, in fact, Ken St. Andre's reaction to the lack of unified mechanics in D&D. He saw the promise in the method of play, and rather quickly made the leap to story-mode-only, but used a "Saving Roll on Luck" mechanic, with difficulty levels. By 4th ed, it was used for saves on all the other stats, too.
Every stat except Charisma has a clear mechanical use in T&T.
Strength limits what weapons you can wield and armor you can wear. Also powers magic in 1.0-5.0. Part of the adds calculation. Also used for fatigue.
Dexterity limits what weapons you can use. Also part of the adds calculation. Limits what level of spells can be cast.
Constitution is used as hit points.
IQ is used to limit which levels of spells can be learned. Dump stat for Warriors.
Charisma is used in resisting certain spells, and has vague implications for awe.
Luck is used for saving rolls versus a lot of stuff, and is part of the adds calculation.
The saving roll mechanic is 2d6 + stat for a target number or higher, with doubles open ending. It was used for just about everything except combat.
Later editions (edition 5.0 and later) add Speed as an option, and (editions 5.5, 7.0, 7.5 ) add Pow/Magic/Kremm for powering spells in place of ST.
The Fantasy Trip: (1981) everything is some number of d6's vs attribute, roll low, or is a flat chance (usually on 1d6, or rarely 2d6) defined by rules. Even combat. Only 3.5 attributes... ST, DX, IQ... and Movement Allowance. Some GM's treated MA as an attribute, others as merely a racial ability.
Classic Traveller: (1977) Attributes take damage, provide modifiers to some skill uses (including weapons), and most rolls are 2d6 roll high, but some skill subsystems are 2d6 roll low, and a few use 3d6. Some rolls are for stat or less on 2 or 3 d6.
It was equally as incoherent as D&D, especially since most skills have their own resolution rules. Combat was a cohesive 2d6 + Skill ±1 for Dex of certain values (by weapon).
I started with Moldvay, then migrated to Traveller... then to TFT, and then to a wide variety; by 1987, I'd been introduced to T&T, using it almost exclusively for solo play. Only later did I get the little brown D&D books.
Rolling for stat or less is so intuitive that, even if not in a game, many gamers will want to do so when a coherent rules mechanic isn't already dominant. But it's always been at odds with the "More is Better, so higher is better" mentality, leading to a variety of roll high approaches (including several in Classic Traveller alone, and in T&T).