I'm not getting your point, Dick, about the D major rather than the D minor - if by D mixolydian you mean the mixolydian that uses the notes that are two semitones above the mixolydian that uses all the white notes on the piano, then the key is D major (or its relative minor, B minor) neither of which has a D minor chord in it. It's nothing to do with modes, just the key.

If you mean however the mixolydian that runs from D to D then it's not in that key at all but in G major (or E minor) - neither of which (again) has a D minor chord.

Or am I missing something?