You are correct, "stoiciocrat" cf "stoichiocracy" is apparently not the correct term. When I learned about this concept about 8 years ago [1], I seem to misremember this to be the word on wikipedia. Apparently the correct spelling is "stochocracy" although the results are very few if I try to look them up today. I was referring to concepts that seem to be more conventionally called sortition (apparently the better term at least in academia according to Google Scholar result counts), demarchy, random ballot...

[1] The history page of "stochocracy" at wikipedia suggests this wasn't my imagination:

https://en.wikipedia.org/w/index.php?title=Stochocracy&actio...

I also remember distinctly the page about the Kleroterion / Cleroterium, which I highly encourage to read:

http://www.alamut.com/subj/artiface/deadMedia/agoraMuseum.ht...

While I still consider myself a "stochocrat", this is but one of my many ideologies.

I believe many different forms of governance are the best depending on the problem class or issue class, or properties of the problems themselves. For example a the problem "choose or design a setup to isolate heavy water from naturally abundant water" will not fare well under sortition, since it requires expertise. On the other hand, problems that have been broken down to such simplicity that anyone could execute it, but that typically suffer from corruption by a minority fare better under sortition: perhaps the selection or patrol routes and the assignment of cops to groups that run the patrol routes, would make it harder to initiate corruption, if you do not know ahead of time who will be your colleague, nor when or where.

Most decisions are not made by governments, but by individuals, and nowadays also computers (or this could perhaps more properly be seen as amplified decisions by programmers).

So if you ask "Do you know if many others share these views, and if they've ever been implemented?" I can not really answer, at least in the past systems have existed where people were raised with the concept of sortition, and all systems indoctrinate the next generation with the importance of itself. If you ask a person on the street what they think about sortition, they will probably not know the word nor the concept, and after explaining mechanically how to do it, they would probably consider it crazy and inapplicable to any problem. I think the number of people wanting to learn more about sortition would increase if not only explained mechanically, but also the properties (good and bad) were explained.

Have they ever been implemented? I consider it an important part of Athenian democracy. Also sortition is alive and well in many systems and places, think of sortition of a jury of peers, think of implicit sortition in mining a block in the Bitcoin system, think of explicit sortition in Algorand, ...

Note the last 2 examples are digital systems: a requirement for sortition to work is that "nearly anyone can satisfy the task" (such as verifying the signatures of transactions, and checking for sufficiency of funds, which any computer could do in principle) as opposed to "design a lock-in amplifier".

I apologize for my incoherent writing style