It's the other way around. You usually had to be rich to become a writer.

Estimates put literacy in the first century CE around 10 percent, although there's a lot of variation when you get into what counts as literacy. There were probably more people who could read than people who could read AND write, and more people who were literate enough to handle household or business accounts than could read philosophical texts.

Literacy takes years of training, and more if you want to be able to produce your own writing. Before widespread public schooling, the vast majority of people couldn't afford the time, let alone the cost to hire and/or support tutors. So most writers of all types of literature were either aristocrats or the slaves or freedmen of aristocrats.

Writers made their money in various ways. Some, like Virgil, received patronage for their writing from even richer people. Others, like Cicero, had their own sources of income from business, inheritance, land ownership, marriage, etc. They didn't normally sell their writing, like modern writers do, and live off of royalties.