Yitz,
I did some further research and Rambam seems to say explicitly (Guide 3;40) that the government may execute people according to circumstantial evidence. This is also Rabbi Bleich’s interpretation (“Contemporary Halachic Problems” v. 2 p.365). On page 362 he also quotes Rav Tzvi Hirsch Chajes (Torat Nevi’im ch. 7) as saying that the government may even execute people according to estimation and that this appears to also be the view of the Chatam Sofer (Responsa Orech Chaim 208). He also quotes Rav Chajes as saying that rebels may even be executed for minor infractions as I previously posted to prevent a breakdown of law and order. This is his slippery slope.
As for injustices, how would you feel if c”v a loved one was murdered and the killer was set free because of lack of certainty such as the absence of witnesses (such as the case of Shimon ben Shatach and the man who chased his fellow into a ruin and came out with a bloody sword)?