I’ve always just thought in terms of the ordinal position of each chord, within the tone center. So:
The flaw in this, is that it doesn’t adapt well as a form of shorthand for chord charts, but it’s dandy for analyzing songs and tone centers.
Outside of weirdness like the blues, I tend to think this way, too. But then I encounter something like the progression of "All of Me," and I ask myself, "Self, what is this madness?"
Every jazz guy (I'm not a jazz guy, but a guy who plays some jazz) I know says the same thing:
<shrugs shoulders> "I just follow the chords."