If the counter bass to any given button was a fifth like you said you could understand in your original question, you'd actually be wasting a button because you already have the fifth right next to the fundemental in the next column :) As stated, the counter bass being a major third makes bass runs and scales a lot easier.

Also, consider, for example, the chord sequence of |G D | Em / |. A nice way to connect these might be to use an F# as the bass note for the D chord so you'll produce a cool descending bass line with the chords - the bass line would be G, F# (counter bass of the D chord), E - without having to go out of position to get the F#.

Another cool use of the counter bass is with an instrument that plays diminished triads on the 7th chord row, like older accordions. I've an 80 bass which has no dedicated diminished chord row. The 7th chord mechanism has been converted to lose a note and leave the three that produce a diminished triad. For example, a C7 chord has the notes C, E, G and Bb. Remove the C and you are left with an E diminished triad. So now you can play the chord with the fundemental C to produce a really sweet C7 sound, or you can play the same chord with the counter bass of C (E) to give an E diminished sound.

I suppose the quick answer to why do accordions have a counter bass is because it's really useful :)