Burnham has never worn a black badge. Not even when she was a convict.
The other cons didn't wear black badges either.
Exactly this. The whole line of reasoning is faulty and grasping at straws...
Ironically, now that you mention it, they might very well be like the Grey Wardens in Dragon Age.
The Grey Wardens (Dragon Age):
“You will guard them and they will hate you for it. Whenever there is not a Blight actively crawling over the surface, humanity will do its best to forget how much they need you. And that's good. We need to stand apart from them, even if they have to push us away to make us do it. That is the only way we can ever make the hard decisions.” ―Kristoff, Commander of the Grey of Orlais during the Blessed Age
[. . .]
Beholden to no temporal lords, they were free to sacrifice villages and fortresses that could not be won, preserving their might for the clashes that truly mattered.
... they acquire whatever resources they can regardless of the method, and employ whomever they must, regardless of their defects of morals or character. They often turn a blind eye to blood magic, deal with demons of the Fade, and do whatever else they must so long as those efforts led to the Blight's end.
... the ranks of the Order are filled with dangerous men and women, some unscrupulous in the extreme, but the Wardens make a point of suspending judgment of any who are effective against the darkspawn.
[. . .]
Grey Wardens are also known to recruit criminals, giving them a chance to redeem themselves by pledging to a good cause.
Clear as day.
...I rest my case.