Given that the FDA is a product of a democratically elected government that is ultimately accountable to the people, while the BBB is a 'non-profit' corporation, accountable to nobody, I think you missed one other important difference.
If I don't like the FDA, I can vote for someone who will fix it. I happen to think that they have, overall, done a far better job than most examples of industry self-regulation.
The head of the BBB doesn't have a monopoly guaranteed by men with rifles and cages.
Do you have any evidence to back up your claim that if you dislike the FDA you can vote someone in who will fix it? I provided pretty compelling evidence that suggests you cannot.
We're all responding to a story of the FDA allowing a big megacorp give us all cancer for literally decades. I think we have a very different definition of the word "good." You can't vote in a better FDA, but you can fire the BBB by simply ignoring them. No system will be perfect, but a better system is one in which you can choose to ignore an unreliable source of information and aren't forced to fund their production of unreliable information.
All the work involved in capturing the FDA demonstrates that the FDA is not a "mandatory BBB", if they really had no teeth there would be no need to devote all these resources to dupe/intimidate/capture them.
Of course the FDA has teeth. But the teeth are only sharp enough to hurt the little fish. The big fish keep on swimming while the FDA takes out the little fish, just like the big fish like it. The FDA and the big fish are partners. Notice how nobody in this situation is partners with you and me.
If I don't like the FDA, I can vote for someone who will fix it. I happen to think that they have, overall, done a far better job than most examples of industry self-regulation.