"The problem I raise here is...what type of human being one ought to BREED, ought to WILL, as more valuable, more worthy of life, more certain of the future". [Nietzsche A 3]

We can see here that Nietzsche almost regarded breeding and willing as synonymous. The application of will to the problem of the 'Higher Man' would surely manifest in a form of Eugenics;"Once the full implications of evolutionary biology are grasped, Eugenics will inevitably become part of the religion of the future, or of whatever complex of sentiments may in the future take the place of organised religion". [J.Huxley, 'Eugenics and Society', 1941]

Without what Nietzsche calls the 'pathos of distance' there can be no overcoming, no contest, no 'agon', no war!A society of Supermen would stagnate, or else break into factions and assert a new Rangordnung which would once again raise the spectre of Eugenics.

I don't think the Overman doctrine is idealistic; the Superman is not to be some spotless Christ-like figure. As Nietzsche writes;
"The word 'SUPERMAN' as designating a type that has turned out supremely well, in antithesis to 'modern' men, to 'good' men, to Christians and other nihilists--a word which in the mouth of a Zarathustra, the DESTROYER of morality, becomes a very thoughtful word--has been understood almost everywhere with perfect innocence in the sense of those values whose antithesis makes its appearance in the figure of Zarathustra: that is to say, as an idealistic type of higher species of man, half 'saint', half 'genius'...He into whose ear I whispered he should look about him for a Cesare Borgia rather than for a Parsifal did not believe his ears". [Nietzsche,'Ecce Homo', 'Books' 1]

I actually think that 'post-Nietzschean' science [e.g., Genetics, Quantum's Uncertainty Principle, and developments in Astronomy] is far more in line with Nietzsche's thinking than the science of his own day was. What is needed today is a 'Quantum Morality', i.e., Nietzscheanism.

Nietzsche makes a strong psychological point when he says that the ever-widening social distances [as in an Aristocratic society] create a widening in the noble individual's 'soul'. In other words his will is strengthened and multiplied. The problem with ANY kind of egalitarianism is that it narrows the 'soul'.