Since the introduction of 5G Release 18, non-terrestrial networks (NTNs) based positioning has garnered significant interest due to its numerous applications, including emergency services, lawful intercept, and charging and tariff services. This release considers single low-earth-orbit (LEO) positioning explicitly for $\textit{location verification}$ purposes, which requires a fairly coarse location estimate. To understand the future trajectory of NTN-based localization in 6G, we first provide a comprehensive overview of the evolution of 3rd Generation Partnership Project (3GPP) localization techniques, with specific emphasis on the current activities in 5G related to NTN location verification. We then delineate the suitability of LEOs for location-based services and emphasize increased interest in LEO-based positioning. In order to provide support for more accurate positioning in 6G using LEOs, we identify two NTN positioning systems that are likely study items for 6G: (i) multi-LEO positioning, and (ii) augmenting single-LEO and multi-LEO setups with Global Navigation Satellite System (GNSS), especially when an insufficient number of GNSS satellites (such as 2) are visible. We evaluate the accuracy of both systems through a 3GPP-compliant simulation study using a Cramér-Rao lower bound (CRLB) analysis. Our findings suggest that NTN technology has significant potential to provide accurate positioning of UEs in scenarios where GNSS signals may be weak or unavailable, but there are technical challenges in accommodating these solutions in 3GPP. We conclude with a discussion on the research landscape and key open problems related to NTN-based positioning.