Sure, that one example does involve only basic manipulation. I wouldn’t say that it’s a problem for a third grader, as while it does involve only basic arithmetic mechanically, it does involve an insight into mathematical relations that a third (or sixth) grader would not generally have. Nor is it exactly a fundamental tenet medicine that doctors must know lest they be terrible doctors, though it is something that can come up that is contrary to intuition (a very important thing for a doctor) that can very well be relevant to medicine. The problem is, instead, something a little more subtle and a lot more significant.
Calculus is the place where college level mathematics begins. Without it, you will not be able to understand any mathematically-focused topic in college: physics, statistics past the first course, linear algebra, computer science past a very basic level, probability, engineering, and anything that has any of these topics as a prerequisite. When they say “some calculus” they basically mean, “we don’t intend for you to use math but we feel that we should have this requirement anyways.” It’s safe to say that someone who only takes only “some calculus” is expressing their intent to never use math again, even in the case of very simple problems like the previous cancer screening one. And if you don’t finish calculus, then you go no further and your math education stops there. It’s the same reason why engineers go further, to linear algebra and differential equations, when most of them won’t ever even touch that stuff: because those basics open the door to further, useful studies in the field.
And as for what doctors actually need math for? Strictly speaking, you can be a doctor without ever using any advanced mathematics for the rest of your life (many are, to the point that it’s a stereotype). It will, however, mean that you lose the ability to have any advanced understanding of physics or data, which are both pretty useful things to understand for a doctor. Those who practice medicine are in a powerful position to be able to make advancements within the medical field, often being capable of doing far more than just researchers who don’t have a medical background. An understanding of statistics and probability at a decent level, and an understanding of computation and software, goes a long way to being able to make sense of information obtained about patients, diseases, and the like (more advanced conclusions than that screening example, but the same kind of information can be obtained from statistical analysis). An understanding of physics helps understand the physics of the human body, and of medical devices which exist or could be created that would be helpful in medicine. All of that comes from topics that are dependent on calculus.
Can you still be successful, perhaps very successful, as a doctor who doesn’t do calculus and does none of that? Absolutely, and many are. Does it mean you shouldn’t learn calculus? Absolutely not. You could be a doctor without knowing history or without having good writing skills either, but I’m sure no one advocates that.
Let me put it another way: you can do higher math without statistics, but you most certainly cannot do higher math without calculus.
