Many thanks for your comments, especially d652482.
You'd probably be swamped with hundreds of requests every day. That's not really so great when you just want to focus on writing.
You are so right. I didn't think about that. I want to focus on research and writing, not on customer service. -). But I realize it is a big part of the job if you have more clients than you can actually work with. I imagine it takes time to service potential clients who never end up paying anyway. That's what I want to avoid, if possible.
At this point I think I'll have someone create a simple, business-like website with my name and contact info and in the meantime I'll work for essay agencies. If someone contacts me directly, I'll work with them using my own website. At this point I can only rely on word of mouth clients, but that should bring more clients.