The point being that you can't be tried for the same crime by different courts at the same time, one court has to take precedence and run the full process to its conclusion. Also the "Not Guilty" plea isn't to say he didn't commit a crime but that they won't admit "Guilty" to what has been proposed as the crime(s) and then be instantly hit with whatever the prosecution wants, i.e. death penalty. The process needs to play out legitimately, and will likely lead to conviction but of crime(s) and with a sentence that is more acceptable to the defence.
If you shoplifted and then are charged with a laundry list of crimes in addition to and more severe than theft, that would result in vastly more severe consequences, you're probably not going to put in a "Guilty" plea for that list.
Sooo... which is it?
But yeah. Obviously America should do this.