You won't gain additional momentum from a loose belt. But a loose belt may give your body the chance to slip out. The impulse you feel from a tight belt will be no less than the impulse you feel from a loose one.
Impulse is constant, its the force that is transferred to your body. This is constant whether or not a belt is worn.
A belt simply takes that force and redistributes it into the seat frame, if the belt is not worn the force is not redistributed by the belt but rather by your head into the overhead bin
The impact is a matter of energy, not of force- if the force applies for more than an instant, you keep accelerating before hitting the ceiling and the energy of the impact will be proportional to the square of your speed. If the belt is tight, you stop at a lower speed (relative to the aircraft) and the energy of the impact (with the belt, in this case) is lower.
It's like debt. If I'm standing on the ground, I'm constantly "making payments" to the ground. If I jump off a cliff, then while I am falling, I stop making payments, so when I next contact the ground, accumulated debt suddenly must be paid.
More precisely, while I am falling, my velocity and consequently my momentum relative to the Earth steadily increases, which is analogous to debt because I will eventually need to restore the situation in which my momentum relative to the Earth is zero (because that is how people live their lives: at rest relative to the surface of the Earth rather than for example in orbit).
In the same way, the passenger in the airplane must eventually go back to being at rest relative to the airplane, but if the seat belt is very loose, "debt" (momentum relative to the airliner) can be accumulated before all the slack has been taken out of the seat belt.
Yes, the total amount of impulse (change in momentum caused by the environment's pushing on the person) integrated over time is the same regardless of how tight the seat belt is, but it matters whether that impulse is spread out over time or comes in one big jolt.
That doesn't make sense. Force is not accumulated by people standing on the ground, so your later analogy does not work at all
The ground opposes the force produced by gravity, there is no accumulation of anything. Gravity accelerates mass when it is pulling it in motion unless there is an equal and opposite force which is the normal force exerted by the ground (or airplane)
If you're harness is loose, or your line is slack without an energy absorber, it'll break your back when you fall!