I put in the different degrees of injury to set the context for the doctor's choice... maybe it takes 5 times as long to save the severely injured person. I didn't mean to imply that the severity of the injury affects the moral calculation.
You're right, this is like the trolley problem. When all 6 people are anonymous, we do the calculation and kill 1 to save 5. When the trolley problem is framed as "push the fat man off the bridge", that's enough personalization to trigger the other part of the brain.
Moral philosophy in general tries to find universal principles whose logical consequences agree with our moral intuition. The OP is saying that we can fix consequentialism by making the moral calculations more complicated. Good luck with that! If moral intuition comes from two different parts of the brain that don't agree with each other, then we can always construct moral dilemmas by framing situations so that they activate one part or another of our brains.
The transplant dilemma is framed in a way that personalizes the healthy young traveler while keeping the other five patients anonymous. This activates the part of our brains that treats people as individuals rather than numbers. There's nothing wrong with the math. We object to the unfairness to the only individual in the story.
This dilemma is usually paired with an example of triage. Here an emergency-room doctor has to choose between saving one severely injured patient or five moderately injured patients. Five lives or one, the numbers are the same, but as long as all six patients are anonymous, it remains a numeric problem, and no one has a problem with the math.
Omega knows that I have no patience for logical paradoxes, and will delegate my decision to a quantum coin-flipper exploiting the Conway-Kochen theorem. Hilarity ensues.