This study addressed the complex problems of selecting a constitutive model to objectively characterize asphalt mixtures and accurately determine their viscoelastic properties, which are influenced by numerous variables. Inaccuracies in model or parameter determination can result in significant discrepancies between the calculated and measured results of the pavement’s structural dynamic response. To address this, the research utilized the physical engineering principles of asphalt pavement structure to perform dynamic modulus tests on three types of high-content rubberized asphalt mixtures (HCRAM) within the surface layer. The research aimed to investigate the influencing factors of the dynamic modulus and establish a comprehensive master curve. This study also critically evaluated the capabilities of three viscoelastic models—the three-parameter solid model, the classical Maxwell model, and the classical Kelvin model—in depicting the dynamic modulus of HCRAM. The findings indicated a negative correlation between the dynamic modulus of the asphalt mixture and temperature, while a positive association exists between the loading frequency and temperature, with the impact of the loading frequency diminishing as the temperature increases. Notably, the three-parameter solid model was identified as the most accurate in describing the viscoelastic properties of the HCRAM. Furthermore, the dynamic response calculations revealed that most indexes in the surface layer’s dynamic response are highest when evaluated using the three-parameter viscoelastic model, underscoring its potential to enhance the pavement performance’s predictive accuracy. This research provides valuable insights into optimizing the material performance and guiding the pavement design and maintenance strategies.