I did say GENERAL rule. And I don't think a mode can be built around a chord.

What would you call the "dominant 7th" if you are using pure mixolydian (White notes, scale starting on G). If the "dominant" is D, i.e. the fifth note of the scale, you surely don't want a D7 chord, which has a sharp in it (a note not in the scale), and neither do you want a G7 instead of a G?

There is also a mode based on the scale and start note of C which is NOT the same as the Major key, though it might look like it. Ionian, I think from memory.