Non-terrestrial networks (NTNs) are considered one of the key enablers in sixth-generation (6G) wireless networks; and with their rapid growth, system-level metrics analysis adds crucial understanding into NTN system performance. Applying stochastic geometry (SG) as a system-level analysis tool in the context of NTN offers novel insights into the network tradeoffs. In this paper, we study and highlight NTN common system-level metrics from three perspectives: NTN platform types, typical communication issues, and application scenarios. In addition to summarizing existing research, we study the best-suited SG models for different platforms and system-level metrics which have not been well studied in the literature. In addition, we showcase NTN-dominated prospective application scenarios. Finally, we carry out a performance analysis of system-level metrics for these applications based on SG models.