A lot of these errors are more in the confusing phrasing or unstated assumptions.
In Monty Hall the ambiguity is around whether the host is randomly opening doors or not.
With the doctor example, they'd normally only face that question when they've ordered a test for someone, so the probability is much higher.
The other classic 'unintuitive' result is the prisoners dilemma, because people have emotional bonds to colleagues and if the break them those emotions can lead to revenge and retribution. These have to be ignored in the classic formulation, but recast it as a drug deal or spy exchange and it makes more sense to people.
At least the way I've heard it, the doctor problem is given as an answer to the question "Why don't we test everyone for HIV/cancer/horrible disease X?" See for example the discussions people had when the American Cancer Society recommended that women with average cancer risk delay their first mammogram to age 45.
It's relevant to that question, but it's also cited regularly as an example of how "even doctors can't do stats".
Which, like most people, they probably can't. But asking someone with lots of experience with a situation, a question about a superficially, but not actually, similar situation adds another level of confusion beyond inability to work out the numbers logically.
In Monty Hall the ambiguity is around whether the host is randomly opening doors or not.
With the doctor example, they'd normally only face that question when they've ordered a test for someone, so the probability is much higher.
The other classic 'unintuitive' result is the prisoners dilemma, because people have emotional bonds to colleagues and if the break them those emotions can lead to revenge and retribution. These have to be ignored in the classic formulation, but recast it as a drug deal or spy exchange and it makes more sense to people.