Interesting proposal!
I think in practice it would devolve into "join my group and vote for me". You'd see people on Twitter trying to convince their followers to join their fan group. Once they got themselves a group of 100, they'd tell followers to join the group of a friend who was ideologically aligned.
You could also form the groups randomly. But assigning every person in the US to a 100-200 person group, arranging for them to meet in person, and letting them socialize enough to know who to pick could be logistically challenging, especially to do in a timely manner. Perhaps if you did it via the internet and used an app such as gather.town to make it feasible to get to know your group. Or perhaps if 100-200 person groups persisted between elections so the overhead of getting to know people was a one-time thing.
All these systems except sortition result in "rule by extraverts", which is arguably undesirable.
An approach that allows for introvert rulers: Let every person nominate whoever they want (including themselves?) Put all those names in a hat, then draw names randomly from the hat for the purpose of sortition. That way if I'm an introvert, and my introvert friend has insightful political opinions, I can nominate them. People would nominate their favorite celebrity a lot, but the flip side of that would be that if you voted for a celebrity whose name was already drawn, your vote is effectively wasted. So there's a bit of an incentive to nominate someone obscure that no one else will nominate. You would probably want to check and see if a person was actually willing to serve before nominating them. (I think it would be unwise to make service in this body mandatory the way jury duty is?)
That one reminds me of liquid democracy, which I believe Glen Weyl has criticized on the basis that it leads to polarization, though I'm not sure why.
