Good question, but first I am NOT a medical doctor, but here is my take from my experience as an electronics maintenance engineer.
MD's would have a degree in some science field, before going on to do their MD it would be at that stage they would do most of the math and statistics.
But in general it would only be in specific area's such as populations and a virus that statistics would be applied. A doctor would however use 'statistical trends' to help lead them to a particular diagnosis of a condition, or to determine area's of risk for a patent, things like having a higher probability of lung cancer if you are a smoker.
But specifically for an individual patient the diagnosis process would be much like what I would do when required to find out what is wrong with a complex system (for me a radar system for example, for them a human system).
They make observations with some understanding of the probability of what could be causing the condition or symptoms that the patient has. Much as we have studied the 'fault modes' of electronic components.
We know the probability that a resistor will probably go open circuit, a capacitor will probably go short circuit, an inductor open circuit and so on, but apart from that we would follow the same procedure as a doctor and employ "Logical Fault Finding", you look at the symptoms and you work back from that to identify the causal factors that would create those symptoms.
So statistics does not really play a big role for a doctor, except in the case you described when a doctor is talking about a group of people and is using statistics to assist in understanding the effects of some condition on a population, or the possible outcome of some condition or procedure.
I expect that a good doctor would have many of the same qualities as a good technician, he/she would have a good knowledge of the fundamental components and how they work, a good knowledge of how the various components interact with each other that makes the overall system work, good logic and reasoning skills and knowledge of diagonstic techniques (such as the 'half split rule' in electronics), and a good overall memory of past examples of the same problems (or fault conditions).
Again in that statistics are only secondary, after you have performed the above you might consider the statistical probably to help you focus on the more probably cause of the observed symptoms.
I would not consider that not being good at statistics would be a reason for me to now go to a particular docotor, but they certainly should have a basic understanding of statistics, science and math as a basic scientific foundation.
Also science in general does not really work much like it is normally depicted on TV or in documentries, science in general is very much a collaboration, so if a scientist doing mechanical or structural engineering (science research) will often collaborate with a mathemetician for help with the formulation and understanding of the math.
Same with physics, Einstein did not come up with his equations by himself, for sure he worked with mathematicians. In the Richard Feynman lectures about fundamental natural laws of nature talks about talking to matheticians about coming up with a math framework for working in multiple dimensions for example.
He makes a joke about that by saying he then asked the math guys to do the same for a single dimension, saying the math guy just said "make n =1", n being the number of dimensions he wanted to work in.
I worked in a civil engineering department at a university does fluid dynamics research, we were working on improving the equations used to determine wave height and frequency/period on bodies of water determined by wind speed and other conditions.
We used to go to lectures by the math department head where he would 'help us' with the mathematics and how best to deal with the data we collected and how those equations worked.
So I would not be to concerned about a medical doctor not being great in statistics, and in conducting research they would seek assistance from statisticians in the design and analysis of trials and research, just as all scientists do.
Thanks, that's an interesting question, and I hope I helped you a bit and was not too all over the place.... !!