Nexus firmware comes straight from google. S3 firmware has to be built to work on the s3, tested, then the carrier gets that firmware, messes around with it, tests it then pushes it to the phone.
Im running 4.4.1 rev1 on my s3 so the answer is to root and use 3rd party firmware instead of samsung's. Basicly make your s3 into a nexus :thumbup: