Because Rapgenius hasn't profited out of the lyrics yet. DMCA focused on monetization of other people's content as one of the main criterias for violation. It doesn't matter whether it's UGC or not.
Case in point: YouTube contains tens of millions of copyright infringing videos. YouTube runs ads on these videos. YouTube was sued for hosting these infringing videos (Viacom v YouTube). YouTube did host the infringing material, and profit from it; that was undisputed. It was found not liable for that infringement because of the DMCA.
As much as people hated it at the time, the DMCA is a pretty reasonable law (which is why the bad guys at various companies are trying to replace it).
If their site had no lyrics, they would have no traffic and as a result no funding.