You can't "Enable management of content types" directly through the library settings on the Site Pages library, but I have just discovered a workaround!
Under Site Actions, choose More Options > Web Part Page > Create. In the Save Location dropdown, choose Site Pages. Voila! You now have a web part page in the Site Pages library.
Note that this still won't add the "web part page" content type to the New dropdown in the library, nor will the "Page Library" command group appear in the Page Ribbon of this web part page, as it does with a Wiki Page in the library.