You are doing it's just that 251's and common scales scream at you on this tune. CST works very well on non functional harmony and modal tunes non of which apply to this tune. Over conventional 251's NO jazz player in the world thinks Dorian Mixolydian Ionian. While the theory is correct in practice its one common scale and chord tones to highlight the chords. This not going against the course method, just using more than one part of it.. Over time we will all just look at a leadsheet and see it for what it is, a guide for playing a tune.
lol They scream at you Chris because of your knowledge and experience and appreciation of chords and chord progressions. For some of us they eventually put their hand up and whisper "I'm over here" for fear of being overlooked forever. I totally agree that seeing the progressions and therefore the common scales is the quickest way there, but when you say "don't think modes do it this way", a way we haven't really covered, it needs fleshing out so that its understood in context of what we've been learning. Not everyone will just get it. I very much welcome your guidance and the benefit of your knowledge but it needs to interact with and build on what we've learned so far rather than be a leap of faith. I know this probably sounds simplistic and ignorant to you but I for one AM VERY ignorant on this stuff and I'm guessing there will be a few other people to whom its not obvious as well. For example, I have no idea what a "key centered" piece really means. To really help us you need to lead us gently in the right direction whilst at the same time maintaining our faith in what we think we've learned so far (or correct us where we're wrong).