The unduly low or high data are commonly regarded as outliers in the classical power geometric operator. However, in many cases, these types of data may be significantly important to the aggregated results. This study aims at expanding the practical application scope of the power geometric operator and then utilizing it to develop a proportional hesitant fuzzy linguistic large-scale group decision-making (LSGDM) model. The extended power geometric (EPG) operator is first introduced, in which these outliers can be distinguished as sufficiently important or "false/biased" data in accordance with the decision-making context. Several useful properties and application characteristics of the EPG operator are highlighted. Subsequently, the proportional hesitant fuzzy linguistic normalized Manhattan distance is proposed, which is a basic concept for the construction of the proportional hesitant fuzzy linguistic extended power geometry (PHFLEPG) operator. Combined with the clustering model for decision-makers, a PHFLEPG-operator-based consensus-reaching approach is provided to simplify and rationalize the decision-making process. Furthermore, the comprehensive LSGDM result is derived by utilizing the PHFLEPG operator. Eventually, a case study on regulatory capacity evaluation for the Civil Aviation Safety Regulatory Authority of China is performed to validate the feasibility and effectiveness of the established LSGDM method.