O Tannenbaum implies a dominant 7th chord at the end of the third measure.

I too rather raise an eyebrow when I read the word "necessary" used in this context. Nothing in music is "necessary".
(Or, from the other point of view, everything in music is necessary).

And as far as "color chords"... well, don't forget that black, white and grey are colours. For me, modal-sounding fourths and fifths (harmonies) are just as colourful as 7ths. They're just a different colour.