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.
For the median American that works full-time, who earns around $56,000, the cost of Spotify is 0.2% of their income.
Obviously everyone is in their own financial situation and for some people that will still be a burden. But for sake of comparison, in 1989, the average cost of cable TV was $18, and that's of course in 1989 dollars. I don't think it's unreasonable to say that $11 for unlimited access to basically all music at any time is a pretty good value for the money.
That said, if it is still onerous to someone, they're obviously not in ideal circumstances and I'm really not gonna fault them for just pirating.
Spotify is not profitable right now; their current revenue doesn't come anywhere close to covering their costs. The only reason they're able to survive is investors holding on to the belief that, some day, it'll be possible to actually make the numbers work.
I don't get why anyone would use the free tier - not being able to choose songs would actually drive me insane, let alone the ads - but $11 monthly for essentially all the music anyone could ever want, plus solid playlists and recommendations, is a perfectly good value for me. Admittedly, I listen to music all the time and it's a pretty big part of my life, so it's an easy sell.
I guess the free tier is still an improvement over radio, but regardless, producing and distributing music has costs, and I'm more than happy to pay for it. Given that Spotify isn't even profitable, having lost about a billion dollars last year, I'm not sure how long this situation will last, but for the time being, I really don't mind it.