It isn't necessary to have a PhD to have an academic career in psychiatry, but if an IMG seriously wants to match into a research track at one of the most competitive programs then yes a PhD is a very good idea, especially if you are coming from an LEDC. Additionally, it will be much easier for someone to establish a research career and get awards if they have a PhD. Actually, depending on the area of the PhD, some may come out of it with few if any publications (though of course the more high quality publications one gets the better) because some fields take years to produce results etc. The fact is that the MD/PhD has become the clearest route to a career as a physician scientists. It isn't the only route but it is certainly the preferred route (infuriatingly as most of the people who state so do not have PhDs themselves...)
I am talking about people who want to have a serious academic career. Having some "spare time to do research" as you put is is not what I call an academic career, and is increasingly difficult for clinical faculty to do with some exceptions (some departments expect everyone to do some research or be involved). There is also nothing stopping one from being a co-investigator in a clinical trial or some other study, but this is a far cry for having a research career - where the typical model is 80/20 research/clinical and you increasingly have to fund your position. If you're not looking to "conquer the world" as you put it, you're not going to get very far in academic.
Only the craziest people do PhDs after residency training, but it does happen (some of my friends have done so). Matt State, who is chair of psych at UCSF did his PhD after residency and fellowship, and stanford has a specific program for residents to do a PhD during residency. I believe Mt Sinai has a 3-year PhD program during psych residency for those who don't already have it as part of their research track, and I know yale has allowed this.
If you just want to dabble in research (which is what it sounds like) it would be a complete waste of time to do a PhD.