As I noted on some other thread a year or two ago, "The Way to Eden" arrived around the same time as the Broadway original cast album of Hair became popular - we had it on 8-track in early 1969 and would often listen to it in the car (my parents were liberal about words being only words and not dangerous in themselves; I was 12). They had recently gone to NY to see the show, and I'd recently seen the episode first-run.
It seems to me self-evident that the episode as filmed - although presumably not in its early "Joanna" version, as mentioned in Gerrold's The World of Star Trek - would not have been made if not for the popularity of the Hair songs at the time.
[In addition to the cast album, versions of at least three songs from Hair were often heard on the radio: "Hair" (the Cowsills), "Easy to Be Hard" (Three Dog Night), and "Good Morning Starshine" (Oliver). Not to mention the number-one hit of 1969: the medley of the show's first and last songs, "Aquarius/Let the Sunshine In" by the Fifth Dimension.]
I've never undertaken a study of (for example) the Hair costumes versus those of Dr. Sevrin's gang, but perhaps one influenced the other there as well.
What irks me about the episode is that it's slapdash in the small details - not only the reversed shot of Kirk watching Adam in the tree in the 4th act, but the three (four?) different pronunciations of Irina's surname, the poorly directed line readings (Adam's "Like he said now..."; the absurd matter-of-factness of Spock declaring "Dr. Sevrin is insane"), the sheer tiredness of the device of having an enemy-of-the-week taking or attempting to take control of the Enterprise, Sevrin's very convoluted ears with no explanation whatever (how would the character have been different with Earth-average ears?), etc.