Sixth-Generation (6G) technologies will revolutionize the wireless ecosystem by enabling the delivery of futuristic services through terrestrial and non-terrestrial transmissions. In this context, the Non-Terrestrial Network (NTN) is growing in importance owing to its capability to deliver services anywhere and anytime and also provide coverage in areas that are unreachable by any conventional Terrestrial Network (TN). The exploitation of the same radio technology could greatly facilitate the integration of NTNs and TNs into a unified wireless system. Since New Radio (NR) is the de facto standard to deliver manifold heterogeneous services in terrestrial wireless systems, 3GPP is investigating new solutions to extend NR to NTNs. In this article, the constraints that NTN features place on NR procedures are investigated by going thoroughly into 3GPP specifications; strengths and weaknesses of the NR technology in enabling typical 6G services on NR-enabled NTNs are identified; and open issues and insights are provided as guidelines to steer future research toward 6G NTNs.