Is it possible to manage content types for the Site Pages library in the browser? If this is not possible, what's the rationale that this can be done in nearly every library and list, but not Site Pages?
No, it is not possible. The OnLoad event of the Advanced Settings page hides the Content Types section with the following code:
this.ContentTypeSection.Visible = this.List.AllowContentTypes && this.List.BaseTemplate != SPListTemplateType.UserInformation && this.List.BaseTemplate != SPListTemplateType.WebPageLibrary;
I do not know the reason why it is protected, but it is definitely explicitly singled out along with the User Information list. That said, you might be able to set ContentTypesEnabled using code. But you will likely be in unsupported territory if you do so. If possible, you may want to consider using Publishing Pages, as those support multiple content types.
I think they want you to use the Enteprise Wiki functionality if you want Content Types and Wiki pages. The Enterprise Wiki functionality uses the Publishing Pages library (unlike ordinary Wiki pages that use the Site Pages library) and the Pages library has Content Type management enabled for the whole Page Layout Publishing architecture to work.