i am using the stock sprint s3 rom 4.4.2 (used to be rooted) and the game works for me.
Where did you get the game? It shouldn't be in the play store (yet) for the s3, I had to copy the apk from my s7 and installed it from my google drive.
the only issue I was getting, was after loging in with my user name and password, I saw a screen with the background but nothing to click on. there's a button that doesn't show up on the s3 near the bottom center. keep clicking (with sound on so you know you click it) and you will get past that prompt for the professor to talk and give you a tour.
Other than that, i have no issues. throwing pokemon balls is a little more challenging on the S3 than the S7, but it runs well enough to play,
Make sure your location setting is on "High Accuracy" and dont have "allow mock locations checked
What Android version did you use?
I'm on 4.4.2 it starts but I get "GPS signal not found", and parts (or all) of the UI are missing.
quick video showing game play -- sound/music doesnt get turned on until later
youtube.com/
watch?v=gvDb1bR3XfA