I think your misgivings are well grounded. I certainly agree with N that the absence of any religious belief systems from which to derive our moral norms requires a complete overhaul in the thinking of unbelievers. However, he seems to think that any 'drive', simply because humans experience it, has no validity above any other, and that 'stronger humans' (usually male in his thought) are the ones who should determine the values that others live by. In such a scenario things could get very messy indeed. No doubt we'd have a society in which the likes of that vile Met Police officer, recently convicted, ran riot, simply because they were 'expressing their drives'.
Yes, his ideas seem to object to religions championing the opposite of biologically driven 'strengths' such as the meek, the humble, the merciful, the compassionate, the poor, the down-trodden etc. A moral outlook that values and encourages these attributes would have to develop for a society to progress in order to balance the 'survival of the fittest' biological mantra, which could lead to outlooks such as that of the Met police officer.
In fact Nietzsche was far from living out these ideas in his own life, except in his dedication to his work. He appears to have been the mildest of men, and a rare self-revelatory quote shows the contradictions in his make-up. Can't remember it exactly, but it goes somethings like this "A few moments of conversation in a railway carriage with simple, ordinary people, and my whole philosophy is in ruins".
I think the only way for an atheistic society to run would be to accept that altruism and general humane behaviour have their roots in evolutionary development. Social groups which cooperate tend to thrive, and develop an evolutionary advantage. The values such societies hold may well coincide exactly with those that many religious denominations hold dear, but without the belief that such values have any supernatural origin.
Agree with your last paragraph but personally I think the supernatural part serves a useful purpose. Religious motivations IRL influence individuals from seeking retribution or power over someone, based on a belief in accountability to a supernatural entity. It prevents a messy situation and helps the individuals involves gain peace of mind because they put accountability and justice in the hands of an infallible entity. I very much doubt accountability to their peers would have the same effect. With the current fallible process of accountability to their peers as the only consequence, would there not be more chance of dissatisfaction with the justice of any outcome, as well as an incentive to calculate the odds of cheating the system, or gamble on no one finding out if they act against laws and moral norms?