No, it's not a free choice. There are established notation conventions designed to ensure that writers write in such a way that readers can read and correctly interpret the music.
Then I would indeed challenge the 'teacher'. If someone has taken something written in A minor and annotated it to say that it was written in C major just because the key signature had no sharps and flats then they were wrong and I'm afraid you were misled. Plain and simple. Here's the acid test - what did it sound like?
I wish I could remember the name of the song.. but.. you know it wasn't just the teacher. It was a pretty popular song that it's been a long time since.. but.. it sounded like A minor. But.. was notated C Maj.
I'll give the possibility that the notation may have been incorrect but.. I mean.. I think his name was Joe or something.. a jazz student of guitar who couldn't really do jack with helping me with my right hand technique on bass.
Here's the big point though. Bro you can sit and talk semantics all you want. If Im absolutely wrong by all means.. let me stand corrected.. however.. I'm not trying to pass a test.
If you adhere to the notes contained within A Aeolian, you are still playing the 5th mode of the Major scale in C major. I've heard more than one musician refer to a song in A minor with a natural minor feel, or D minor with a Dorian feel, as being in the key of C major.
That works for me. I'm not here to argue about that nor do I really get too hung up on semantics.
What I can tell you.. is that if you actually sit down and play modes enough and put some thought into them here and there they kind of become ingrained like any other aspect of correlation between tones and they can be used to enhance your music. You CAN effectively change the key of a song by means of a modal shift and a chord change. You can opt to do it momentarily.. you can opt to use it as a transition into a bridge, or chorus, or a improv jam. I.e. I can not unsee that a C7 chord is composed of notes from the 5th mode of the major scale in the key of F. And really it's just that one single note differentiating between that and Cmaj7. The flat 7. But changing that one single note is all it takes. And I've gathered that through utilization of my fretboard, and my ears more than any other aspect of learning music. If anything.. having that muscle memory and ear development down made putting the other pieces together regarding my grasp of theory just a bit easier.
If I didn't have that understanding I probably wouldn't be able to keep up with musicians in an improv situation.