I think the only way some could possibly be convinced of Brad's guilt would require a video of Brad committing the murder.
IF Brad takes a plea for second degree, stands up in court, and admits guilt, it will be argued he has no choice and he's just trying to save the rest of his life and is willing to sacrifice the next 7 to 10 years in state prison for that. (I'm not saying it makes sense, that's just the rationalization that will be used by some). Brad is not going to stand up and admit guilt for a first degree conviction; he would at least take his chances with a trial.
OR IF Brad turns down the plea deal and instead allows the case to go to trial #2 and then IF another jury concludes that Brad is guilty of killing his wife, with all evidence allowed in, and a jury of 12 still convicts him, some will argue that is "unfair" and "corrupt," regardless of the evidence. (Note, not all will say or believe that, but some will).
Thus, for some subset of people, there is no way Brad's guilt could ever be proven to them given the options available, not even by Brad saying the word, "guilty."