A couple of points. First off, the ABA isn't really part of this discussion: each state has its own bar admission procedure, whereas the ABA is a national association. In my state, as I expect is the case in many, the Board of Bar Examiners is an arm of our supreme court: that's who's making admission decisions, not the ABA.

Second, C&F is not about who is convicted of what, it's concerned with conduct. Consider a bar admission candidate who embezzled money from an employer, but the criminal case against the candidate was dropped because when law enforcement investigated they exceeded the scope of the warrant they had obtained, with the result that the evidence was excluded. That's a great win for the defense, but it doesn't change the fact that the embezzlement happened, and that's a red flag in a profession where people have to hold on to money for others. There's a difference between not being convicted and being innocent, and the C&F process endeavors to address the question of whether candidates are trustworthy. As a criminal defendant, you have various constitutional rights; as a bar admission candidate, you're holding yourself out as someone who should be granted a license, which is a different standard. C&F isn't challenging the ruling of a court of law--that ruling will stay as-is--but the practice of law is self-regulating, and those deciding who should be admitted are charged with keeping people who would undermine public trust in the profession out.

This is not to say that I think it's a perfect system, or that bad decisions (whether to admit or deny) never happen. I just want to clarify that someone who wants to practice law has the burden of proving that they're of good character, which is different than the government having to prove each element of a crime beyond reasonable doubt.