It’s such a staple of writing advice that it’s almost impossible to pinpoint where the quote comes from. When you start looking into writing quotes and statements, every successful author seems to have written some variation on the theme. Writers write. They don’t talk about writing, or think about writing, or wish they’d written — they write.
It’s simple, straightforward, and unarguable.
Someone who thinks about writing, or talks about writing, or wishes they were writing isn’t, well, writing. So they’re not a writer.
The trouble with general advice is that there’s no nuance; no context. What does “writers write” actually mean, in the bigger picture of life?