There's a well known human tendency to stop thinking about a problem and go along with someone else's leadership if they assume an attitude of leadership.
So in this case a fully qualified medical doctor was attending to the patient and the CSM may well have felt that he did not have to provide initiative any longer for the patient treatment. Therefore he was no longer in a mindset of proposing solutions and equipment for the "sick passenger" problem, and not in a mindset to mention the on board epi-pen and similar things. He was probably concentrating on managing the cabin in the aircraft, including such things as the strongly-trained need for security and safety management of outside and cabin doors.
Given the passenger had been given two epi-pen shots by then, I also have to ask how much a third would have helped. I'm not sure the inquest has addressed that question yet.
The doctor was clearly in a bad position as well - probably not entirely aware of the extent of the patient's active and severe allergies, minimal and unfamiliar emergency kit, trained in general medicine but not to a high level of emergency medicine which is a specialty in its own right, and so on.
Serious breathing problems can also be insidious in emergency situations. The patient is breathing, but clearly insufficiently, and is (in retrospect) in immediate danger of death because their condition is rapidly and inexorably deteriorating. However, second by second, the situation does not get very much worse and so the patient may appear to be in a serious yet stabilised condition, until they die. This is also how people die of serious asthma attacks despite active medical attention unless someone follows immediate procedures for very active intervention and has the right equipment to hand.