People keep trying to do it, but I think there's a rather substantial challenge in putting a jetpack in a cover shooter. In a cover shooter you evade damage by hiding behind your friendly neighborhood chest high wall. If you are jet packing, you aren't in cover, so you'll be shot and punished for using your jetpack.

This is worse with modern games built around hitscan or nearly hitscan weapons, since you can't meaningfully evade fire in the air. Most games being built around slower-aiming thumbsticks doesn't help, since in order to be able to fly and shoot effectively, you need to keep a relatively low angular velocity between you and your targets. This translates into slow flight when shooting, and even less ability to evade.

In other words there's a reason that the granddaddy of jetpack shooters, Tribes, is built for keyboard and mouse, and combined super-fast player movement with relatively low speed projectiles. Also why every gameplay video of Anthem I've seen has the enemy dudes just sort of standing there and not shooting very much.