This has all gotten a lot more confusing than it needs to be--first off, forget about modes, they are not really part of the answer to this question--
The major diatonic scale really harmonizes with I, VI, and V7 chords--if you leave the 7th out, the harmony looses it's dominant harmony quality, and it becomes ambiguous to the ear, because it doesn't have the unresolved, dominant sound, and it can sound like you have changed key--Generally, the dominant tension still is present in the melody, though, so you can get away without it in the chordal accompaniment--
The V7 chord has four notes in it, and is called the dominant chord--the fourth note adds the unresolved feeling that we are accustomed to in our western diatonic music, which bounces back and forth between a sort of open or unresolved sound that harmonizes with the Fifth, and a closed or resolved sound that harmonizes with the Fundamental.
The tension, or unresolved feeling comes from the fact that the V7chord contains a diminished chord (G7 has B-D-F, a B-diminished chord, and if you add the G to the top of the chord, you will hear a major second interval F-G, making it a very dissonant chord)..
The chord C6 contains C-E-G-A and G7 has D-F-G-B, which accounts for the whole scale, and the oldest diatonic major melodies that we still use are among the tunes used for Morris dancing. and they generally just move back and forth between the Tonic and Dominant chords--
Hope this makes things clearer, although one never knows--