Answer: This question provides a nice opportunity to say again that notation is not meant to be a visual form of tape recording, but is more like an architect's plan for a building. The architect's plan does not actually depict every nail and bracket; some details are left up to the builder.

But in this case the architect was worried about the nails being put in the wrong place and decided to include them in the drawing.

To escape that metaphor, it sounds to me like this arranger has decided to notate "swing" literally. If you see pairs of eighth notes in 4/4 you will normally assume that they are played evenly, as they appear on the page. But in jazz it would be more natural to "swing" them, i.e., to make the first of each pair a little longer than the second, remembering the advice of Duke Ellington: "It Don't Mean A Thing (If It Ain't Got That Swing)."

It might have been enough just to write the music in the easy-to-read way, as eighth notes, and maybe write the instruction "swing" at the beginning. But this arranger wasn't sure the reader would know what that meant, and has decided to try to write out the swing rhythm.

Here's a line written normally,

Here are the same notes written as triplets, trying to give the impression of swing:

"Swing" actually has an equivalent in baroque and early classical practice, called "notes inégales" (unequal notes), and in fact the practice of swinging pairs of notes may be even older than that. The music was written the plain way, both for ease of writing and ease of reading, but in practice performers might lengthen the first note of each pair.

As for using sextuplets instead of triplets, the implication is that the "4/4" has a feeling of two beats in the bar, each one with the value of a half note. That suggests that the piece really should have used a "2/2" key signature instead of "4/4," but there's a lot of flexibility allowed here, and essentially the sextuplet is the same as two triplets. Could be that the arranger just thought it would save work, writing two sextuplets in a measure instead of 4 triplets.

How can you tell when you have not just notated swing but a "real" 12/8, i.e. 4 beats each divided in 3? The tipoff would be numerous places in which the triplet included 3 equal notes rather than a long-short pair. Here's a true 12/8 melody from Bach:

Even if I had notated that in 4/4 with triplets it would still be 12/8 in effect. The groups of three make it clear that this is more than swing, it's really based on beats that divide in 3.