Graves is free to try this week and I found his autoattack to be, as a friend described it to me, 'Very snappy'.

I can't confirm how many champions have issues here, but I find it greatly improves the ability to last hit and also just makes the game feel much smoother in general if a champions "windup" animation for their autoattack is very quick. Graves' attack is a good example of what I am talking about.

I often play Twitch and his animation seems to take an extra few microseconds that makes his attack feel slightly delayed.

I see that as well.

I wonder if it's only the two of us.

I know the "windups" are somewhat intentional for example almost no "ADC" has one yet almost all the "supports" do. Its all tied into what your character does most often if he/she is meant to be aa reliant they will have smooth motions if they're heavily spell reliant then likely it will be slower.
In my opinion:
Worst delay: are Janna and Anivia
"Jerkiest": Muay Thai Lee sin and Graves

Trying one bump.

I think that Riot is aware of this and is managing the attack animations on purpose as part of game balance. Janna, Anivia, and Karthus all have sluggish auto-attacks, but none of them are supposed to rely on those heavily.

Indeed. Part of each champion's gameplay comes in his auto attack. To a degree, it's what determines what champions can farm better than others and alters the way you play them during laning phase. For example, Graves farms better than Twitch, but Twitch pokes and harasses better than Graves (opinion alert). This applies across the game and actually is fundamental to certain champ's gameplay, such as Skarner, for example.