Suppose we did not separate these things. In that case, for an entire group to have A, that implies that some ancestor of that entire group had A. But this contradicts the premise that the gene makes people less likely to reproduce.

It works if you accept the idea that genes which are fit in some situations are less fit in others. In that case you could have a person who developed this mutation of feeling suicidal when they feel useless, but it doesn't "activate" because they aren't in a situation where they feel useless. But once a whole group has this gene, then your theory of propagation through siblings/cousins is more plausible.