Displaying written lyrics does not require a royalty payment to anyone.

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.

... Why would the lyrics service allow that?

The issue isn't the storage, it's the copyright holders