Why not select political office holders at random in the same way jury selection is done?

The only culture to have a working government - the strongest and longest lasting government in the recorded history of the world - were the Iroquois. They established their Constitutional Republic in 909CE, and it lasted until the European invasion.

They had full equality. Theirs was a democracy by consensus. That meant that the representatives were true public servants. They had no autonomous political power. But those who were elected were the WISEST among them. Wisdom was to their culture what money is to ours.

Because jurists are not supposed to know much about the cases they will sit on, but political office holders are expected to know enough about their jobs to serve competently.

I think jury duty should be voluntary like running for public office.

Some people would be literally incapable of performing those duties to an acceptable standard

Iroquois also had FULL participation by it's constituents.