In current practice, organ transplant recipients are typically old people who die shortly after receiving the transplant. The problem is still interesting; but you have to impose some artificial restrictions.

Sure, it's just a thought experiment, like trolley problems. I've seen it used in arguments against consequentialism/utilitarianism, but I'm not sure how many of utilitarians bite this bullet (I guess it depends what type of consequentialist/utilitarian you are).