Might it be that a history of mental illness prevent a job in psychiatry?

It could... BUT...: Firstly, "mental illness" is ubiquitous! it's a very broad term! next, there's a huge stigma to overcome and tremendous ignorance, in general, about it. Having been through a psychological trauma; having managed yourself out of a chemical addiction or a clinical depression... Can give you rich insights, possibly allowing you to be a superior clinician! it all depends on what type? ...How severe?

I agree with Dr. : Lane. If an individual is significantly impaired then becoming a psychiatrist might be like the blind leading the blind. If an individual never suffered trauma, emotional pain or dysfunctional relationships - then they might not have much of a base from which to build insight or empathy. What is essential is that a provider has dealt with their own demons & can use them to be better a therapist.

Psychiatry: Mental illness well controlled and in remission should ideally not preclude a job anywhere but sadly thats not true.

Shouldn't: Ongoing symptoms that impair one's ability to full the role obligation of employment as a psychiatrist, such as impairments in working relationships, could cause problems. A history, meaning stable symptoms and that person has attended to their own self-care, is itself a good sign in my view!