Juries have a lot of "professional supervision." In the Common Law system, the judge restricts who can serve on the jury, determines the relevant law, tells the jury what specific question of fact they are deciding, controls the evidence shown to the jury, does the sentencing, and more. My impression is that the non-Common Law systems that use juries give them even less discretion. So when we have citizen-volunteers, we get good results only by very carefully hemming them in with professionals.

You can't supervise the executive in the same way. By definition, the executive is the part of the government in control of the coercive apparatus. If the nominal executives aren't able to give orders to the military without the approval of some other body, then the nominal executives aren't really in charge; they're just constitutional decoration, like the modern British monarchy, or the Presidium of the USSR.