The author never suggested that jazz cannot have improvisation. The issue is whether that improvisation strays far from the melody. The author thinks if it does, it shouldn’t be classified as jazz.

The author never suggested that the later music wasn’t heavily influenced by jazz. What the author cares about is the level of improvisation. We want to weaken the author’s assumption that straying far from the melody should prevent a music from being jazz.

The identity of the performers has no clear impact on whether music should be classified as jazz. Jazz musicians can perform non-jazz music.

The author never assumed that only jazz contains improvisation. The issue is whether straying far from the melody should prevent a music from being classified as jazz.

This suggests that if we are to classify the music based on similarities/differences, there’s no more reasonable option than classifying it as jazz. Other types of music are more different than early jazz is, even if the later music’s improvisation strays further from the melody.