Attunement, as Peter points out, is basically about restraining magic item swapping, which is seen as abuse. You can't just slip a ring of regeneration on whoever is injured, you can't all share boots of levitation to get up a cliff one by one. You have to take the time to attune each person to the item before using it. This makes having just one copy of an item an actual drawback, instead of a trivial detail.
Some of this certainly occurred in OD&D and AD&D, but it just wasn't seen as abuse yet. Why can't the party share a pair of boots of levitation? The only difference attunment makes, in that case, is triggering multiple wandering monster checks instead of just one. But something like putting a ring of regeneration on an unconscious and dying comrade would seem more like abuse. What stopped players from doing that?
It was the forgotten feature of D&D magic: the magic name or word of command.
Read Magic: The means by which the incantations on an item or scroll are read. Without such a spell or similar device magic is unintelligible to even a Magic-User.These days, people forget that Read Magic was primarily aimed at controlling the use of scrolls and magic items, not at controlling spell books. Aside from magic potions and maybe, maybe armor and weapons, everything was supposed to have a word of command. If you didn't know the magic word, you couldn't use the item.
This doesn't completely prevent magic item swaps, but it does mean that the person using the magic item has to make a conscious decisions to activate the magic. So, a party can share a ring of regeneration, but you can't use it to revive a recently-slain character, because that character can't speak the magic word. You also can't use it if otherwise unable to speak.
The iffy part was the magic weapons and armor. Armor was pretty simplistic at first, just offering defense bonuses when worn. So, too, were most magic weapons, with the exception of magic swords and the dwarven warhammer. You could go with what Peter called "splitting the effects" in these cases, the sword would just be a drab +1 magic sword with no powers, the armor is +1 plate, unless the user knows the magic word to activate its other powers.
Of course, magic swords had another feature back then that was sort of like attunement, but less gamey: Ego. Magic swords are intelligent and can withhold their powers or even betray their owner if they don't like the way they are treated or are jealous of other magic items. This could lead to far more interesting play than "cuddling with your sword for an hour" to activate it.
Both the magic word and the ego of swords are game world features as well as limitations. Attunement has a nominal game-world explanation (you meditate or perform some trivial ritual to link yourself to the magic item,) but this is hand-waved. The important part, for modern games, is the limitation of abuse that attunement provides, not the way it affects the character's actions in the game world. It's less entertaining.
Why did gamers switch to less-interesting game rules? Was it purely because of how complicated the older rules were? I could see this for the Ego rules, but not the magic word of command rule.
I think we should bring back magic items with names.