Then time goes on and no one pays the premiums after death; however, no one claims the life insurance pay out, also.
Is the life insurance company still obligated to pay out the money even though, for example, a year has gone by from the day of your death before the beneficiary claims it and no premiums were paid during that one year?
Yes, the life insurance company is still obligated to pay the death benefit of the life insurance policy.
As long as the policy was "In Force" on the date of your death, even if it's five years later, they still have to pay out the claim.
Premiums are not due on a life insurance policy after the insured person has died.
But, you still need to make sure your family members know where your life insurance is, and how to contact the life insurance company upon your death.