He shouldn't say it because, having received the sacrament of extreme unction, we Catholics presume that he died as one of the faithful departed. In case you do not know what a sacrament is, "a sacrament is an outward sign instituted by Christ to give grace".
If you have faith in the power of the sacrament, then you know he died a member of the faithful departed. If you have no faith in the power of the sacrament, then continue being logical I guess.
Not even in Novus Ordo textbooks have I read such a thing.
Even THEY say that there is such a thing as receiving sacraments unworthily and without any effect. They even give precisely this same example of the lifelong sinner who confesses on his deathbed, but since he never confessed in life when he was healthy, he didn't know how to do a good confession at the hour of death and consequently did not profit from the sacrament.
Go back to reading the catechism why don't you.
Oh wait, it's "heretical" and all you need is a Denzinger.