Unfortunately the only way to get around this is to have agreements in place with independent artists for the site that you could then provide to the RIAA when they come knocking. If you post music by artists signed to a label it's out of the artists' hand even if you attempt to put something in place directly with the artist. The RIAA is aggressive and well funded. Their entire objective is to make sure artists get paid whenever their music is played regardless of whether the site is free/no ads.
EDIT: Spotify, Apple Music, Pandora, etc do not provide APIs to embed their players into other sites without some sort of licensing agreement in place.
Or couldn't you make a front end that pulls audio from Spotify or some other paid service? I want to see ideas like this flourish and support artists.