If you only pitch to publishers you know accept unagented material, you'll still have a big list of publishers for an agent to contact because most publishers (especially the big ones) will not accept unagented material. In theory, it could work out to try without an agent first.
However, an agent is likely to look at your rejections from publishers as an indication that the book won't sell, and that could hamper your efforts. You have to tell the agent that you already pitched directly to those publishers.
It's a crapshoot either way.