"The main point is that the gene's effect must cause copies of that gene to become more frequent in the next generation. "
Pah! I disagree. The gene's effect must not cause copies of that gene to become less frequent in the next generation is the very best you can hope for. More accurate still is that in conflict and cooperation with other genes in the organism the genes effect must not afford less of a chance of being given a free ride.
Isn't the vast majority of DNA just genes that don't do any harm? Evolution is not the thrusting, carving, shaping force depicted so often, it is (as usual) the work of a small (say 20% (a guess)) of genes and the rest just don't compete so get a free ride.