You are correct. For instance, traditionally, in barbershop vocal music, singers are trained to deliberately deviate from 12 equal, towards an ever-shifting kind of just intonation, in order to maximize the extent to which the voices blend. Auto tune, on the other hand, just tunes things straight to 12 equal. Melodyne fares a little bit better in that it lets you tune to custom microtonal scales, or fudge things a little bit here and there, etc. Interestingly, Logic Pro X has a "Hermode Tuning algorithm" that will basically do the dynamic adjustment toward just intonation for you, but it only works for MIDI instruments and not auto tune as far as I know.
Autotune software has different parameters available to it. These include pitch correction speed, how close a singer has to be to the note in order to start/stop pitch-correction, which pitches to correct for.
The T-Pain effect, which is the autotune sound you’re probably thinking of, cranks most of those parameters all the way up in order to get to that robot voice: “instantly lock the vocals to one of these set pitches, if the singer goes lower than X, immediately switch to the next lower pitch in the set.” More subtle usage makes for a performance that is more in tune overall but keeps much more of the vocalist’s expression and pitch variation intact. Its goal usually is to not be noticed.
I don’t think I understand your question (edit: about it being a bug), so I won’t attempt to answer it directly, but maybe the above info is helpful in thinking about it.
I'm assuming, but I want to check with you all: does Autotune always "correct" to the exact center of the note? I assume the answer is probably Yes.
If so, that's a bug, is it not?