I'm a physician, with an undergrad in physics. My research is very applied and very math heavy. When applying to med schools, I asked a couple very senior mathematicians at my college as to whether I should take a stats class. They assured me I would not need a stats class.
The first time I got a biostats lecture in medical school, I genuinely questioned their advice. By the third class, I was convinced I knew nothing about math. Luckily our biochem prof was a physics major and assured me I didn't need to worry about the stats.
Somewhere in residency, I realized I really had been getting all the stats questions right. But not because I did anything like the other residents. They have all these crazy stories and mnemonics to keep track of what kind of problem they're looking at. They seriously try to classify their way out of basic math. They classify. Everything.
I now agree with my profs. I didn't need a stats class. But the stats education in medical school seems designed to convince physicians that they need statisticians. Which is probably a coup for biostatisticians, but it's a damn shame for physicians trying to break into research outside major academic centers.
Irrelevant to your main point - I am surprised you didn't have at least an intro to probability and statistics, and later a statistical mechanics course as part of a physics undergrad curriculum.
Relevant to your main point - I have worked with many clinicians. Most of them have had a fairly tenuous grasp of the statistics they were working with, which matches your experience in training I think. A couple had much better understanding, but had driven that education themselves.
It is true that depending on how statistical mechanics is taught, it can be difficult to connect to other statistical reasoning (the distributions are "nice", you have a gazillion measurements, and your population sampling doesn't tend to have bias issues, etc.)
The first time I got a biostats lecture in medical school, I genuinely questioned their advice. By the third class, I was convinced I knew nothing about math. Luckily our biochem prof was a physics major and assured me I didn't need to worry about the stats.
Somewhere in residency, I realized I really had been getting all the stats questions right. But not because I did anything like the other residents. They have all these crazy stories and mnemonics to keep track of what kind of problem they're looking at. They seriously try to classify their way out of basic math. They classify. Everything.
I now agree with my profs. I didn't need a stats class. But the stats education in medical school seems designed to convince physicians that they need statisticians. Which is probably a coup for biostatisticians, but it's a damn shame for physicians trying to break into research outside major academic centers.