I think you aren’t factoring in that focus groups are lead by people that are not stupid.
At best, a focus group is directed by well-intentioned people who may not know every pertinent correlation value needed to ensure a representative sample of the public. The conventional wisdom is that focus groups are typically used and designed by people who are somewhat out of touch, and this isn’t entirely false.
At worst, a focus group is directed by people with an interest in showing a certain outcome to a certain set of people. Using the earlier example, AIPAC lobbyists might want to downplay the political damage caused by the Iran War, and can alter candidate selection and discussion framing in order to do so when given authority over study design.
Moreover, participants often want to please the experimenters, and will, consciously or otherwise, speak differently depending on what they think they want.
Put simply, you end up with all the same issues as polling. Potential for bias, design error, and unreliable subjects.
Found a typo: