Reading the 5E PHB, it seems to me like one of the differences between older (as in pre-WotC) D&D and newer D&D is that character abilities more often manifest as “specific beats general” rules. For example: monk class feature: Stillness of Mind: starting at 7th level, you can use your action to end one effect on yourself that is causing you to be charmed or frightened. This is, of course, explicitly discussed as a design principle (page 7), and I think the same language was in 4E as well, though I do not have those books conveniently available to check at the moment.
There are some of these sorts of things in earlier D&D also, though not nearly as many. The paladin’s lay on hands ability is one such power, as is turn undead, and, perhaps, spell casting in general. But they were not nearly so prevalent, and the few classes that made heavy use of this sort of design (as did the AD&D monk and bard) felt different, and maybe even a bit off, like they didn’t fit the system quite as well as classes like the fighter, thief, or magic-user. Spells are the standard bundles of reality distortion available in earlier D&Ds, and while they do certainly increase PC play complexity (especially for those classes that need to make spell preparation decisions also), spells are a bit more cleanly separated from the underlying game engine, compared to the myriad discretionary class features present in the newer editions, and also are usually resources that must be spent, as opposed to options that may continually come into play.
As a matter of game play, such “specific beats general” abilities are cognitively heavier than the choices necessary when playing most classes in older editions. They are part of a catalog that must be kept in active memory. You need to remember that you have Stillness of Mind available as an option when you need to end a charmed or frightened effect. Is this qualitatively different than the equipment inventories that develop with medium to high level characters in older D&D? I am not sure. In any case, equipment inventories are also an aspect of characters in newer D&Ds, so at the very least, when comparing complexities, the comparison is A compared to A + B.
I can’t help but think that the card design paradigm of Magic: The Gathering contributed to this trend. Even when considering alternative system approaches before the onset of WotC D&D, most games seem to focus less on little bundles of rules that accrete to PCs as they develop, and more on a fixed collection of measurements that improve (attack tables, skill ratings, and so forth). For example, the attributes, abilities, and backgrounds in Vampire: The Masquerade are mostly all on the character sheet for all characters to begin with. Characters develop vertically more than horizontally. This is not necessarily meant to be condemnatory, nor is it any kind of ironclad principle (as I am sure there are plenty of exceptions), but it does seem like drift in design sensibilities.