Let's look at the actual requirements at scouting.org. It says, "Play an appropriate video game with a friend for one hour." That means one hour, once. Furthermore, that requirement is one of nine, only six of which must be fulfilled in order to earn the Video Games pin.
As a scout leader, I can tell you that the Boy Scouts of America does not promote spending a lot of time playing video games. We do, however, recognize that video games are a normal part of a kid's life experience. So why not teach our kids how to evaluate systems and games from every aspect? Why not teach them how to think critically about selecting and playing video games?
There is no specific mention in the BSA guidelines about first-person shooters or any other kind of game. The requirement is to understand the ESRB rating system, to be able to explain it, and to be able to make good choices about selecting games. The guidelines don't even prescribe what it means to make good choices; rather, they assume that the boys and their parents are intelligent enough to make their own decisions.
I am a software developer, lifelong gamer, and dad of a scout.