I'm concerned about a jury system of eligible voters because I fear that might create a deep state that's even worse than the one we have. No juror would truly know the government.

The only way that that kind of system would work is to give the randomly selected person the kind of powers that the president has: "all public officials exist solely at the discretion of the specific electee that is currently serving". this would also mean you wouldn't be allowed to have independent bodies, and that entire agencies would have to fall under the purview of a specific electee seat.