Recently, scrap tire rubber-modified asphalt binders and pavements have been the preferred choice of state DOTs and parties involved due to the desirable engineering, as well as economic and environmental impacts. Rheological and mechanical properties of rubber modifications have been the main focus of researchers for the last couple of decades. This paper investigates the rutting potential, fatigue cracking resistance, and continuous performance grade (CPG) changes of waste tire rubber-modified, original, and aged asphalt binders. The CPG of asphalt binders is determined at high, intermediate, and low temperatures. A Delta T Critical comparison of the binder was carried out to establish a relationship between measured parameters. Linear amplitude sweep (LAS) tests at equi-stiffness temperatures were conducted to discover the fatigue life of all binders while the multiple stress creep recovery test is performed to assess the high-temperature rutting performance of asphalt binders as per the Superpave performance grading system at accepted regional (58 °C) as well as high PG temperatures. In addition, parallel-plate geometry and concentric cylinder geometry were used with the Multiple Stress Creep Recovery (MSCR) test to discover the impact of discrete particles available in crumb/ground tire rubber-modified asphalt binders as per standards. The results show that rubber modifications improved the base binder’s rutting resistance and continuous PGs without adversely affecting the fatigue cracking resistance. Based on the mathematical expressions developed, 2.71%, 7.82%, 12.94%, and 18.05% (by weight of binder), GTR modifications improved the high PG of the modified binders one, two, three, and four grade bumps, respectively. Similar linear correlations with R2 0.872 and 0.6 were established for continuous low and intermediate PGs, respectively. MSCR test results revealed that both 9% and 20% GTR modifications were achieved to enhance the H-grade traffic level of the original binder to E-grade.