The advent of next-generation ultra-reliable and low-latency communications (xURLLC) presents stringent and unprecedented requirements for key performance indicators (KPIs). As a disruptive technology, non-orthogonal multiple access (NOMA) harbors the potential to fulfill these stringent KPIs essential for xURLLC. However, the immaturity of research on the tail distributions of these KPIs significantly impedes the application of NOMA to xURLLC. Stochastic network calculus (SNC), as a potent methodology, is leveraged to provide dependable theoretical insights into tail distribution analysis and statistical QoS provisioning (SQP). In this article, we develop a NOMA-assisted uplink xURLLC network architecture that incorporates an SNC-based SQP theoretical framework (SNC-SQP) to support tail distribution analysis in terms of delay, age-of-information (AoI), and reliability. Based on SNC-SQP, an SQP-driven power optimization problem is proposed to minimize transmit power while guaranteeing xURLLC's KPIs on delay, AoI, reliability, and power consumption. Extensive simulations validate our proposed theoretical framework and demonstrate that the proposed power allocation scheme significantly reduces uplink transmit power and outperforms conventional schemes in terms of SQP performance.