It boils down to the project model. Asp.net MVC only works with project files (not the folder approach), so you really have to build it to publish it (which can be using the publish feature and copying All in there - including the bin folder).
You probably could get around it by adding some automation that invokes the compiler, but its just designed for you to upload the assembly for the .cs files you had. Related to this approach would be to include continuous integration to your efforts, or maybe just borrow the piece that takes care of doing the build.
Note: I'd argue that the folder based model for asp.net has less time around than the project based one i.e. since the first versions of asp.net project file based model was the one used, when vs 2005 was introduced they changed to the folder based model and then added the project file one again. Check the link for info on the project file model.