I believe the line you quote is perfectly correct English. I would punctuate it as
"Why she had to go, I don't know--she wouldn't say."
"Why she had to go" is correct grammar if you interpret it as "Her reason for leaving."
"Her reason for leaving, I don't know--she wouldn't say."
"I don't know her reason for leaving--she wouldn't say."
I agree that it _could_ also be bad English, if you imagine that it is something like "Now why she hadda done go? For why she do that thing?" But not in this song, because the other words of the song are in standard English.
And, yes, song lyrics are unreliable. Songwriters often bend grammar for the sake of the rhythm and rhyme. Furthermore, they sometimes write poetry that evokes a mood without having a clear logical meaning.
The Beatles range all over the spectrum. Consider:
I am the eggman
They are the eggmen
I am the walrus
Goo goo g' joob!