The 'icly promoted' phenomenon is generally a field promotion for one reason or another. In Rylee's case, it was the captain putting in a good word with CentCom to allow Rylee to be promoted to Head of Security.
I honestly don't remember Lucina's rise to CMO as clearly, but I believe it happened in a similar manner when the CMO slot opened up one shift and she got the Captain to approve her promotion- for her demonstrated skills in medicine.
Rylee has long since stepped down as HoS due to the stress of being in a head role getting to her, but Lucina, despite her temperament, remains in her role. If she proves herself to be unfit for the non-technical aspects of her job, I will not protest to a demotion just as IC as the promotion she originally gained.
Honestly, I think all head characters should be promoted icly; they would have to actually demonstrate their skill with the department and have positive relations with at the very least the other heads. This is actually far more rigorous a screening than taking the time offscreen to 'go to med school'.