Just as the title says: I don't want to have to manage fonts on my render machines. I just want to embed the necessary fonts in the AE project file. However, I'm not able to figure out how to do this.
You can not embed fonts in AE. An After Effects project only contains links to assets and a script that tells the render engine what to do with the layers. If your animations are set up correctly you may be able to either create the type in Illustrator and create outlines or rasterize type in Photoshop, or create shapes from your type in AE using the Layer>Create Shapes From Text option so that you don't need the fonts.