The concept of citizenship is important to keep in mind when this practice comes up. Only a small minority of Athenians had citizenship even when the democracy was at its height in the 5th century, so the pool of potential officeholders was much more narrow than it would be today if something similar were tried.
You may recall that Andrew Jackson thought that any citizen should be able to perform the duties of public office and that a lottery system was legitimate in his view. Before the 20th century most offices required no specialized knowledge on the part of the officeholder.
I’m not sure whether a lottery system would put the brakes on the growth of the State. It could prevent the creation of a separate class of people who view the State as the way to wealth; that would certainly be helpful.