Modified asphalt binders are still considered important in asphalt pavement. However, the comprehensive use of various modifiers is limited due to storage stability issues. Moreover, there is a scarcity of detailed analyses regarding the degree of separation for asphalt binders among each method despite the utilization of various methods to assess the storage stability of binders. Therefore, a comprehensive analysis was conducted to assess the storage stability of asphalt binder modified with a crumb rubber modifier (CRM) and styrene–isoprene–styrene (SIS), utilizing five evaluation factors following the ASTM D7173 guidelines based on four mixing methods (A: high-shear mixing method, B: low-speed agitating method, C: high-shear mixing method + low mixing method, D: low-speed agitating method + low mixing method). To produce the modified asphalt binder, the proportions of the CRM were 5% and 10% for each binder, and 10% SIS was added to all binders. The results in this study convey that (1) the addition of the modifier led to an increase in G*/sin δ with different mixing methods, but using mixing methods (C and D) for a relatively long time resulted in a lower G*/sin δ, indicating suboptimal performance; (2) through the multiple stress creep recovery (MSCR), rheological properties of Jnr and % rec exhibited trends similar to G*/sin δ evaluation, highlighting an improved elastic recovery with a higher modifier content; (3) storage stability assessment revealed consistent trends in high-shear mixing groups (A and C), while low-speed mixing groups (B and D) exhibited an elevated separation index (SI), suggesting a sensitivity to modification conditions; (4) evaluation using the MSCR method indicated that % rec with a 3.2 kPa load is effective for the sensitive assessment of binder storage stability and Jnr showed a limited sensitivity across varying loads, advocating for % rec for precise evaluation; and (5) despite permitting various tests, achieving consistent results remains challenging. Future research should explore diverse modifiers and optimal evaluation methods to enhance knowledge of binder behavior and separation dynamics.