I can't accept too much applause, OlBlueEyes. One of the reasons I follow OCBRs so closely is that I live in a medium-size market on the West Coast where most of these shows won't play except years from now as community theater. (I'm two hours from Los Angeles, so I do see some things there, including AMELIE.)
As for rhyming, you are correct that near-rhymes have been in use throughout the history of the English language. (Different languages have different customs: Japanese, for example, doesn't use rhyming as a rule, but is very big on alliteration.) For one thing, in poetry that is meant to be read, primarily, paired rhymes like home/come "rhyme" to the eye, if not to the ear. For another, imperfect rhymes feel more "relaxed" and may better fit the contemplative mood of many poems.
In the theater, however, where lyrics must be heard over an orchestra, the rumbling of a turntable, the tapping of the chorus and God only knows what else, perfect rhymes were the norm from the early 20th Century to 1970 or thereabouts. Unconsciously, the theatergoer knows that if she hears "time" at the end of a line, the next line will end with "mime" or "rhyme", not "mind", "thine" or "Hammerstein". This aids her comprehension. Here is one of the great lyrics of all time:
"Here's to the ladies who lunch.
Everybody laugh.
Lounging in their caftans and planning a brunch.
On their own behalf.
Off to the gym,
Then to a fitting,
Claiming they're fat.
And looking grim
'Cause they've been sitting
Choosing a hat.
Does anyone still wear a hat?
I'll drink to that."
I love the above example (from COMPANY, 1970, lyrics by Stephen Sondheim) because it illustrates another rhyming principle: perfect rhymes should highlight (for the ear just as I did above for the eye) the most important words in each line.
What's even more astonishing is that Sondheim repeats the same complex rhyming pattern in TWO MORE choruses! And in doing so allows Joanne to discuss the psychology and philosophy of her social class along with other complex themes. The same content might well be lost in the theater without equivalent perfect rhymes. (People keep quoting wonderful lines from GROUNDHOG DAY in this thread, but I didn't hear a one of them when I first listened to the recording.)
My remarks here apply largely to theater music. Pop music, on the other hand, is designed to be played repeatedly, on the radio and, once-upon-a-time, on the record player; so it has never mattered whether one heard every word on the first hearing. A good hook that allows the listener to remember the song when he gets to the record store is enough. (There are a number of great pop lyricists, of course, and most of us would put Dylan, Springsteen, Henley and Mitchell alongside the best of Broadway wordsmiths.)
And Broadway audiences today are more conditioned by pop than theater music; meanwhile, songs are amplified loudly and clearly. So whether perfect rhyming really matters as much nowadays is a matter for debate. But it's an aspect of craft that we old-timers appreciate.