I would reword your statement to say "Since all chords are organized collections of tones, all chords can be associated with a scale within the modern diatonic & chromatic systems of harmony. Major triads, for example, can be associated with the major scale since both collections contain a root, major 3rd and fifth. Although there are many scales which align themselves with a major triad, the major scale is a more well-known choice and therefore a more digestible association for beginning instrumentalists."
Although there are many theories in music and a whole host of nomenclature and syntax that exist, it all boils down to the same set of 12 tones (in modern Western diatonic & chromatic harmony, that is.) I have found that the easiest way to chords, arpeggios and improvisation to students is to associate the chords with common scales. For example:
CHORD => SHARED TONES => ASSOCIATED SCALE(S)
C Major Triad => 1, 3, 5 => Major, Lydian
C Minor Triad => 1, b3, 5 => Natural Minor (Aeolian), Dorian
C Diminished Triad => 1, b3, b5 => Whole/Half Diminished, Half/Whole Diminished
C Augmented Triad => 1, 3, #5 => Augmented, Whole Tone, 3rd Mode Melodic Minor
C Dominant 7 => 1, 3, 5, b7 => Mixolydian, 5th Mode Harmonic Minor, Super Locrian
These are just a few examples. Every scale contains hosts of possible chord combinations, just as every 3 or 4-note chord is contained within several scales. These simply serve as mechanisms to associate chords with a set tonality and provide a degree of functionality for beginning & intermediate students of music. I hope this helps!