Writers come from all kinds of backgrounds and many different levels of education. Those things aren’t what make a writer good or bad.
The ability to clearly communicate a thought or idea and to take the reader from the real world into the world of the writer’s imagination is what makes a good writer.
I know one writer whose English wasn’t the best. Her publisher suggested she NOT try to polish her skills because her own way of speaking, broken English, faults, and all, was part of what made her writing so memorable.
Okay, that’s the exception, not the rule. But you get my point. Communication skills are more important than English proficiency.