Avatars are meant to be controlled by the user, not by you. So yes, in a way, you’re being forced to use the Gravatar service. But remember, it gives the user the ability to use the same avatar anywhere, and you can always restrict the display of a gravatar based on content ratings (G, PG, PG-13, R).
Gravatar is a hosted service, which is why there’s no “Upload Avatar” section in the profile.
You say you “can’t use a plugin,” but really that’s the only way you can add features. If you want to use something other than Gravatar, you’ll need to load a plug-in to support it. There are a few plug-ins that support local avatars:
Otherwise, I recommend you educate your users on what Gravatars are and how to use them.