Purchasing a song on iTunes does not grant you public performance rights.
Yes it does.
But you must pay IF you do. An Artist cannot stop you from playing the recorded music in a performance, it's part of the deal. Just like Chrissie Hynde cannot stop Rush Limbaugh from using Ohio as his show theme music BUT he must pay the royalty fees to do so. She can complain about it, she doesn't have to accept the royalty check from it BUT she can't stop him from playing it in a performance nor can the publisher etc.