Sortition like this is an interesting idea, but you have forgotten that people who meet those criteria can be emotionally disturbed, mentally compromised, in a coma, physically unable to perform.

Also, what level of a criminal record? Only Felonies (i.e. "serious crimes") and above? Misdemeanors? Even a traffic ticket?

I'll also point out that not all HS diplomas are equal, HS in a few states like NY, Cali, Michigan, are far more rigorous/difficult than schools in say Alabama or Mississippi.

I'm reminded of efforts that most people put forth in limiting suffrage, any criteria you come up with tends to be very blind.

I've personally thought that Sortition is a great idea, but I thought using a system more like Jury duty would be better. Grab your random candidates and then have them interviewed by a few bipartisan lawyers and judges to make sure they're not obviously horrific people.