It was sometimes stated in various AD&D books that you could do some sort of an 'ability check', but it was never clearly defined as a standard mechanic
It certainly wasn't. Not only did different sub-systems use different rolls (d20, d20 vs a matrix, d6, %), but different stats gave different bonuses for the same value (18 CON? +4, if you were a fighter, otherwise +2), 16 STR +1, 16 DEX -2, &c and had different checks CON? Make a percentile 'system shock' roll. STR? d6 to force open a door or % to bend bars DEX? d20 + reaction bonus, maybe? Then there's roll d20 or 3d6 under your stat (or my favorite, roll high on d20 without going over the stat). Consistency was not high on the list of classic-D&D virtues.
5e doesn't really have that one page. It talks about different types of checks in a few places. Skills are presented as non-core rules and their explanation is actually pretty obtuse if you go and just read it cold. In fact they're almost not explained, though anyone who knows modernish D&D and understands the core mechanic will 'get it'. 4e is much more concise and explicit in this sense.
4e strove for relative clarity, while in 5e clarity would almost be superfluous beyond the key explanation of the core resolution mechanic, which is a sentence, not a page. Along the lines of: the player describes an action, the DM decides if a roll is required (if so the player and/or DM might roll) and describes the results of the action.
That's really the bottom line across the board, everything else is window dressing. The rules say this or that, which may or may not be clear or broken or even workable, and you have some stuff on your character sheet that the DM may or may not take into consideration or even look at, but ultimately the DM is told up-front, to do whatever he wants, and that fulfills 5e's promise of DM empowerment, inclusion, modularity, and being all D&Ds to all D&Ders. Which is all, really, very much in keeping with the vision EGG laid out in the 1e DMG, just with a bit less obfuscation.
I find this to be an issue with 5e, so many of its core systems are presented as 'options' that later on when they talk about game process they have a hard time formulating an overall vision of how play proceeds at the table. Thus really needed expositions like 4e's page 42 simply don't exist AT ALL in 5e, instead we get scattered around the books some suggestions that aren't very well cohered together.
Play proceeds at the table in the basic cycle of DM description, player action, DM ruling, optional dice rolling, and back to DM description.
Its worth revisiting WHY 4e doesn't have this structure, a point system effectively. Its because it leads to heavy spamming. Its the same reason you can't just take the same power more than once, the designers wanted each PC to have a wide repertoire of tricks. In a 4e combat each non-at-will power use is unique, you haven't done it before, and won't do it again in that situation.
It was an anti-boredom provision, like that, yes. It also limited the impact of a broken power slipping through the cracks. A traditional or neo- Vancian or spontaneous caster that gets his hands on even one broken spell can cast it a lot, multiplying the impact of that spell-design oversight.