Fiber lasers have been widely used in the industrial and scientific fields due to their advantages of high conversion efficiency, simple thermal management, and consistent stability. High brightness and high-power fiber lasers are affected by stimulated Raman scattering and transverse mode instability, which limits the power scaling of fiber lasers. Therefore, there are only a few researches achieving a 10 kW-level fiber laser system by laser diode direct pumping or tandem pumping. In this work, we demonstrate an all-fiber laser amplifier based on home-made low numerical aperture (NA) fiber pumped by 976 nm laser diodes. When the signal light is input to the gain fiber with a minimum bending diameter of 12 cm, the beam quality factor M ²is about 1.72. The onset of transverse mode instability (TMI) is observed at 2467 W output power, accompanied by beam quality degradation. In order to suppress the onset of TMI, the minimum bending diameter of the gain fiber is changed from 12 cm to 20 cm. And the signal light is input into the gain fiber with a bending diameter of 28 cm. Benefiting from this operation, the fiber laser amplifier achieves maximum output power of 10.53 kW with an optical-to-optical efficiency of 74.04%, and there is no TMI onset observed. However, increasing bending diameter inevitably leads the beam quality to degrade. At the maximum output power, the beam quality factor M ²is 2.88. To the best of our knowledge, this is the highest optical-to-optical efficiency and the best beam quality in 10 kW-level laser diodes pumping fiber lasers. Generally, it is believed that reducing bending diameter can suppress TMI by increasing high-order mode loss. However, this rule is not applicable to few-mode fiber lasers. A larger bending diameter leads more high-order modes to be contained in the signal light instead of leaking into the cladding area. Thus, a higher output and poor beam quality are obtained. Also, it is believed that tightly coiled fiber can make mode coupling easier and trigger off TMI, which results in a positive correlation between the TMI threshold and bending diameter. Low NA fibers are very sensitive to bending, and reducing the bend diameter to control the beam quality will result in lower efficiency and a lower TMI threshold. Therefore, although producing a 10 kW-level fiber laser is simple, maintaining good beam quality in the power scaling process is still a challenge. The results of this study will be a valuable reference for high power fiber laser design.