Punch-Drunk Love is one of my favorite romantic films! Adam Sandler is a damned talented man, don't get me wrong. That's why I'm even more okay with criticizing his more standard fare. The dude can do better.
I think the key to understanding "how to do the FPS better" is that the first-person model doesn't only have to exist for shooting. It's a very potent set-up because it's "virtual", and limits your perceptions to the scope of the character you're controlling. You could make a game that involves guns but is not strictly a shooter, and therefore has the freedom to give you a certain context into what surrounds your actions.
As much as I love Buddhism in my video games, I feel like your idea might feel too preachy to your average player. But how about this for a set-up? The game starts off as a "sports" shooter game – say, paintball, or something of the sort – and you're a person with a couple bucks and a passion for the game. Ultimately, you get to a point where you're up against people with much better equipment than you, and the game becomes almost a test of masochism: how many times will you try before you give up? But there's a whole other world out there, offering you opportunities to make a quick buck. And whether you pick a live of military service or a life of crime, you're put back into a situation with guns, only now the stakes are infinitely higher. There are no second chances.
A game like that could really play with character development, both by looking at the people who serve along with you, and by making enemies something more than faceless soldiers. Give you the impression, perhaps, that the only people fighting these fights are people who don't have the idle money to play paintball all their lives, and that you're being sent to war against yourself, against people who are pretty much just like you. Except that sometimes, you're fighting for the rich people who want more, and sometimes you're fighting as the underdog, against a group of greedy people who are willing to use violence to get what they want.
Of course, if you're looking to avoid that kind of morality, you can always stick to playing paintball. But you're aware of the fact that even there, you haven't quite escaped from the world as thoroughly as you may have liked...
I know nothing about paintball forgive me