Graphics drivers contain competitive optimizations that make a significant difference in the performance of the hardware. Since these optimizations are both costly (in R&D) and allow for the vendors GPU to perform better than a competitors, they do not share these optimizations (that a competitor may or may not use to improve their own driver). And since people pay ultimately for performance, not hardware, the driver is just as important as the silicon.
Your second point is correct, that open source drivers do get better support. AMD has better support on Linux than NVIDIA because it has an officially supported open source driver (AMDGPU). However, the support they receive from open-sourcing the drivers is not always the R&D required to gain a competitive performance advantage, limiting the competitiveness of an open source driver (at least compared to proprietary).
Your second point is correct, that open source drivers do get better support. AMD has better support on Linux than NVIDIA because it has an officially supported open source driver (AMDGPU). However, the support they receive from open-sourcing the drivers is not always the R&D required to gain a competitive performance advantage, limiting the competitiveness of an open source driver (at least compared to proprietary).