Yep. In the US, it’s the venue that is responsible for the license. Technically, even playing a radio station in a public space requires that license, which is why most stores that have background music pay for Muzak or some similar service which bundles all the licensing costs along with the music stream itself. The license covers not just pre-recorded music but live music. I remember reading an account of a cover band which switched to playing originals because their venue didn’t have the licensing for them to play covers.