Great point, P!
It is a rather "primitive" idea that attributes success to the gods. Perhaps it is better called a fallacy. Although the primitive book of Job saw through this.
However, whether he was a believer in the Gods or not, Polybius firmly believed that religion had a very favorable effect upon the character and virtue of the citizens of the Roman Republic.
Histories, Book VI, Chapter 56:
"The sphere in which the Roman commonwealth seems to me to show its superiority most decisively is that of religious belief.
Here we find that the very phenomenon which among other peoples is regarded as a subject for reproach, namely superstition, is actually the element which holds the Roman state together. These matters are treated with such solemnity and introduced so frequently both into public and into private life that nothing could exceed them in importance.
Many people may find this astonishing, but my own view is that the Romans have adopted these practices for the sake of the common people. This approach might not have been necessary had it ever been possible to form a state composed entirely of wise men. But as the masses are always fickle, filled with lawless desires, unreasoning anger and violent passions, they can only be restrained by mysterious terrors or other dramatizations of the subject.
For this reason I believe that the ancients were by no means acting foolishly or haphazardly when they introduced to the people various notions concerning the gods and the belief in the punishments of Hades, but rather that the moderns are foolish and take greater risks in rejecting them.
At any rate, the result is that among the Greeks, apart from anything else, men who hold public office cannot be trusted with the safekeeping of so much as a single talent, even if they have ten accountants and as many seals and twice as many witnesses, whereas among the Romans their magistrates handle large sums of money and scrupulously perform their duty because they have given their word on oath.
Among other nations it is a rare phenomenon to find a man who keeps his hands off public funds and whose record is clean in this respect, while among the Romans it is quite the exception to find a man who had been detected in such conduct."
When he describes such conduct, it should be kept in mind that Polybius was a Greek himself. :-)