I'm very skeptical sortition has any chance of success. A major belief of the liberal revolutions that brought about modern democracy was meritocracy, that an army or state bureaucracy should be run by the most competent person regardless of birth, rank, or status. When you're sending an army off to battle, having it be run by some random guy you dragged in off the street isn't really any better than having it be run by the king's inbred cousin.
Even if you limited eligibility with the goal of ensuring competency (say, by only allowing someone with a college degree to hold office), you could easily wind up with a situation where someone who has spent their life becoming a master engineer suddenly has to run the foreign ministry. Theres a reason that in spite of all the classical LARPing both the French and American revolutionaries loved to do, neither group ever seriously considered it as a means of running a modern state. If anything, I could see a more moderate form of technocracy taking root somewhere in the Western world. Say, for example, that rather than a civilian leader choosing who runs the army, the army itself selects the person they deem most qualified for the role.
I'm surprised you don't see more examples of collective executives in 19th century history. It seems to me like a reasonable way to limit the power of corrupt or incompetent executives, as well as letting them divide their efforts during times of crisis. It may be common knowledge on here, but for about 15 years between the American Revolution and the Constitution, my home state of PA operated this way. Each county got one representative on the executive council, and they chose a President of Pennsylvania. This was probably the primary inspiration for the French Directory (whose failure I suspect probably turned a lot of people off from this style of government).
For America specifically, I'm also surprised the New England town meeting form of local government never made it outside that region. Given the westward settlement of a lot of New Englanders, as well as the isolation and desire for local control common in the Old West, I could see town-level direct democracy growing popular across the Great Plains. A small 19th century American town is practically the perfect environment for direct democracy to thrive, and perhaps you could see modern anarchists talking favorably about American towns that run their affairs locally and collectively.