No source that I'm aware of provides a public API, leaving tools like beets to haphazardly scrape a handful of websites[1].
[1]: https://beets.readthedocs.io/en/v1.6.0/plugins/lyrics.html
It's really unfortunate that there isn't a single, high-quality reference source (with an API) for song lyrics, the way there is for artist/album metadata (MusicBrainz) or cover art (Cover Art Archive).
I can't remember if it was them or another service (edit: I'm pretty sure it was actually https://www.lyricfind.com ) that used to have a free licensing tier if your lyrics site included their ads, which were for a scam ring tone subscription service that would trick users into a ringtone "subscription" billed to their mobile carrier. The ads promised MIDI ringtones on every lyric page, but usually they didn't have them for more obscure songs.
Does Genius not serve this purpose? https://docs.genius.com/#songs-h2
That's why LyricsGenius uses an HTML scraper to extract the lyrics[1], and why other tools (like beets) have data quality issues.
[1]: https://lyricsgenius.readthedocs.io/en/master/how_it_works.h...
No source that I'm aware of provides a public API