According to people smarter than me about this stuff, I did not "turn off" DTMF, I only changed it from Hardware to Software, which should still do the same thing, and should not affect whether CWCID works or doesn't work.
To use your analogy, If I had a camera which had optical (hardware) zoom and digital (software) zoom, and I switched from optical to digital, I should still be able to zoom in or out, just the quality might be different. But it should work at least.
At best, it is a "design" defect.
YMMV