Apple is known for its attention to detail, but this revelation takes that to a whole new level. Instead of using less-than-perfect lyric services to provide lyrics alongside its music streams, Apple Music has "a team of people listening to music and transcribing the lyrics."
The full quote from a WIRED interview with Apple Music chief Oliver Schusser:
In probably the most bizarre example of Apple spending to get the experience right, Schusser says that Apple has “a team of people listening to music and transcribing the lyrics” to ensure they're accurate enough for Apple Music’s new time synced lyrics feature; “we don’t get them from the usual sites.”
Here's an idea, employ a team of people to contact songwriters and music publishers asking for the lyrics and all of the underlying metadata.