We would be hundred of years behind where we are today, and I have already explained why.
But let me explain it again.
If you conquer a group of people with military might, there is always a certain level of resentment and civil unrest that finally leads to rebellion.
If a people willingly submit to a new idea, the risk of rebellion is very limited.
Where nations like Rome failed to defeat the people of Europe as a whole and unify them under a common language and government, religion was able to do just that.
If religion never happened, the different factions of Europe would have been waring for centuries after Rome fell. Without religion we may have never had a Greece, or a Rome.
There would have been no reason for the crusades to happen, so the exchange of cultures and ideas between the muslims and christians would have never happened.
Instead of math being brought to Europe by the returning crusaders, it would have taken centuries to slowly work its way into europe.
Reading and writing that was promoted by monasteries and monks would not have fallen to the wayside.
These examples are just from the middle ages. Without religion to unify people, would we have even made it to the middle ages?
Why was mankind on the planet for 100,000 years, then all of a sudden there was a growth on society in the past 6,000 years? What about the other 94,000 years? What happened for society not to develop?
Why did humanity all of a sudden develop over the past 4,000 - 8,000 years?
Why didn't we have cities and government 50,000 years ago?
What changed between 100,000 years ago, and 6,000 years ago?
The development of an organized religion. Religion is what brought mankind out of the stone age and flourished ideas.
1) The idea that military spread of ideas and technology is the only method is laughable. Case in point, the Silk Road.
2) Taking your example of Rome, in only two places were religious integration even needed - with the various peoples on the Germanic frontiers, and with Jews and Christians. In the case of Jews and Christians, the initial circumstances that led to strained tensions between the Romans and the 'natives' was religious in nature anyway. Without religion as a cause, there would have been no need for religion as a cure. And in the case of the Germanic tribes, the divide was more cultural than religious, which in any case was so great that most attempts to integrate Germans into the Empire was a long-term failure.
3) Please explain how without religion there would be no Greece.
4) Correlation does not equal causation. There are several other factors that you are not taking into account - development of language, development of agriculture and domestication, technological development, etc. In any case, I don't see any evidence to show that religion was a recent invention. And an invention is exactly what it is.
Most of your Roman historians were also the elite who were holding priesthoods in the state religion. While religion wasn't the sole motivation, it certainly played its part.
Saying that a Roman leader held a priesthood is utterly worthless, since the positions were predominantly political in nature. As a proxy for actual belief, it is useless.
And your claim that religion didn't have much effect on the Roman people is patently false. Their calendar was built around religions observances, and sacred places were located all over the place, usually water sources.
Maybe that was a bad choice of words. Yes, they probably had nearly as many religious holidays as work days. But again, those were predominantly political in nature, and were created by ambitious aediles in the Republic or early Empire, when the aedileship still mattered, or by Emperors. I reiterate what I said before; I believe for many Romanized citizens, religion was purely a means to prosperity rather than an end. The Romans were very pragmatic people, I have no problem with the idea of them taking a Pascal's Wager type approach to their belief.
A small example that demonstrates this is the use of the civil service by Constantine and Constantius II to spread Christianity, and then by Julian to try spread paganism. Julian died before any of his policies really got going, but the actions of his cousin and uncle before him speaks volumes about the importance of belief in the Roman Empire.