This the reason I prefer to explain modes as I did. If you choose to view each mode as a seperate centricity then all the other rules of tonal harmony have to be tossed or reinterpreted accordingly. Trying to keep 7 different sets of tonal relationships in your head is pretty cumbersome.
The only system I know that does work well for explaining modes as centricities is set theory, which is also a little cumbersome for answering a quick question like "What is a mode".
I was taught modes in much the way you are describing them way: Dorian = Minor #6, Mixolydian = major b7, etc. I viewed them as seperate and distinct scales and had no clue of how the related to each other, or how they connected with anything else in my musical knowledge.
It wasn't until I read an interview with Alan Holdsworth on how he first taught himself modes (supposedly by playing the major scale in several different variations) that I even knew the modes were related, let along that they fit neatly on top tonal theory.
In my view, since most of us start from a largely tonal background, modes should be explained in a way that relates them to this.
As for various peices and songs being in certain modes, given a score you can make a compelling argument for either. Is a song in Phrygian because it uses a b2, or is in Minor and using a Neopolitan chord?
Unless its in 12 tone or a non-western tonal system, you can distill almost everything into common practice tonal harmony. That's probably why it stuck around as long as it did.