The writers almost certainly do get a cut. Musixmatch does not own the copyright to any of the lyrics, and as such, negotiated with the copyright holders in order to be allowed to store, display, and sell access to the lyrics. This almost certainly involved some amount of money changing hands.
Spotify is paying money for access to the lyrics and using that as a feature in their product. A chunk of that money is almost certainly going back to the actual writers.