I love the idea - there are some defacto situations that might arise i.e. Theocracies and Meritocracies. But the idea is fascinating.
I don't think it is that hard either - it is fundamentally the usurpation of power by a scholar class. I think the reality is that an Iatracratic state is more of a flavour of a Bureaucratic state - with the Doctors having the power because, hello - the not only have the respect of the people they care for, but they also have the army on their side because they stitch them up!
I think you'd want to look at how a highly trained minority group can integrate itself into a society and be respected rather than feared. I did a weird old piece about how Phoenician settlers could eventually become the Pharaohs of Egypt (see here) - and I think having a large number of charitable doctors might be the ticket. The charity works get them respect, whilst they're slowly working themselves into powerful positions. Canny doctors start using the respect they earn to dominate their families (which could include scribes, merchants, etc) and eventually become the advisory class to the Pharaoh.
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If you want a pure Iatracracy - I don't know how feasible it is, but having a ruler that relies on the health of their people for power doesn't seem that removed from the rule of the village elder, or clan leader. I think if you have some sort of horrible disease in a moderately advanced area, you've got the chance of the doctors effectively getting the reward for saving many people. It is a bit handwavy, but perhaps you have something like cholera - with a priesthood that revers water, and a medical class that has cottoned on to the vector of cholera early. Priesthood is wrecked. It isn't hard to see how the idea that the Gods should be revered for saving them seems bonkers, when the Priests all die.
Cue rebuilding, respected doctors essentially lead the community, and the King/Army relies on their support for the support of the people. If the King is gone, I could see level-headed doctors try and co-ordinate with each other, effectively building a flat system, and appointing one of their number to co-ordinate everything, leading to a hierarchy.