I have to disagree with this for a variety of reasons. going to a PhD program just to increase your fellowship match chances is a bad bad idea. Also, if you're genuinely interested in a PhD program, your final goals are usually to be a physician-scientist and you have to understand that this is becoming extremely difficult because of funding. Signing up for a PhD program is a big decision and I never recommend it to med students unless they know exactly why they're doing it and have a clear objective in mind. Doing it because "it looks good and it will probably help you in the future" is a bad life decision.
I am not encouraging OP. To apply for a competitive fellowship, you have to be competitive. Research makes a CV more competitive. In the context of OP, I outlined that his options would include not doing any research and going for it, doing some research during residency, taking time off and doing a lot of research. My advice would be that if s/he were to do a lot of research, then doing it under the umbrella of a structured program with a degree (MS, PhD, MSci, whatever) at the end would be more optimal in my opinion. I agree that going for a commitment such as a PhD strictly to guarantee a good fellowship is a bad idea because it is overkill and will result in a lot of misery along the way. But that being said, people won't know how much they or dislike something until they try it. I can give teens of examples for that.
My personal experience is that I took 2 years off and did research not because I wanted to, but because I really did not know what am I doing with my life. The experience I gained from that was invaluable - publications, but more so, interacting with people and facets of the research world that I would have never gotten otherwise. I still wish I could have consolidated my experience into something such as a masters degree.
Also from personal experience and public knowledge, people with MD/PhD who apply get substantially better interviews and match at highly ranked academic places as compared to others.
Bottom line, my point previously was that more research is better. If OP really wants to invest in it, then a PhD is an option to consider. There are other options that will get a good fellowship though.
My point now is that it is not the worst idea in the world.