This one is actually out of their hands. Lyrics aren't free sadly and they have to pay for API calls. It's fucking stupid but the labels are the ones at fault here.
Unless there's some agreement / licensing thing prohibiting it, and considering that lyrics don't change, they should be able to do some caching for a total of 1 API call per song
You may get lucky and have a band that doesn't care or won't notice your operation, but Spotify has music from the big labels and they do care and they will notice.
I assume they are not paying for the lyrics, but for the access to the api. The lyrics are also timed to the music and the service they use might do that for them. So, like you say, serving lyrics data costs very little, but that is not what they pay for.
And to add, I don’t really know anything about how this works behind the scenes.
It very much is and Spotify would definitely get sued if they weren't paying. I got a cease and desist for an app I made about a decade ago for this very thing