Someone trained to be a medic wouldn't be trained as much as a doctor. And just because you can patch up a wound doesn't mean you know how the body works. I can apply band-aids, but I don't know the process the body goes through or how the band-aid helps it. And a medic wouldn't need to know, most likely.
In fact, some of the skills you listed would be useless to even a doctor. Xenobiology? Botany? Is he going to treat plants from a foreign world for measles?
And in fact, your bio says:
Sune learned basic first aid, and the use of homeopathic healing techniques from the people who found him. While on the scout ship he spent time with the ship's medic to familiarize himself with Yamataian medicine. This is gave him the idea later to join the service, he is proficient in Medical skill.
So, while I'm not going to say what anyone writes is an exhaustive account, there is no mention of science... at all. And I don't think there's anything in your career so far where you could have gained said skills. So, even if it was possible or even the norm for Star Army Medics to have such skills (which I doubt) you shouldn't.
And if Medicine and Science covers all that, just what is the point of having a skill "Biology"? Could I have taken Science and Medical and gotten everything in biology plus surgery skills?
As for the new approach, it specifically points out medics have first aid skill, which again is leagues below a doctor. And everything else you learned was from a bunch of spacers, not the scientific type. It even defines what medical skill is, which is what you gave yourself under Medical and Science: first aid. Not exactly a scientific background.
I don't mind you having gotten the transfer, because screwing up badly is realistic, but Sune should look at the equations and DNA analysis Darnell hands him and kind of stare at it and have no idea what it means. He simply doesn't have the background.