They probably changed it because they couldn't figure out how to stage it. In the original German text, as well as in the production I saw in German and the actually good translation of the play by Joel Agee they strangle him to death, though it's not explicit. Here's what it says in the Agee translation (It's pretty much identical to the German original). Keep in mind that Anton Schell is actually called Alfred Ill.
Ill walks slowly down into the lane of silent men. At its far end, the Gymnast stands facing him. Ill stops, turns around, sees the lane close mercilessly in on him, sinks on his knees. Without a sound, the lane turns into a cluster of bodies that swells and then slowly crouches down. Silence. Enter newspaper reporters downstage left. The lights go on.
So I suppose that means he could also have been beaten to death.
When I saw the musical he was strangled, though I didn't find it awkward. The German production I saw was incredibly strange (it made the Broadway production seem like hyper-realism) and they just kind of stood around him from what I recall and he exited the stage. I liked the musical more than the play, probably because I understood 100% of what they were saying instead of just about 50% :) but the English translation I find better dramatically than the book to the musical.