Considering how many of the Roman writers and politicians emphasized old Roman virtues/values, it doesn't seem unlikely to me that, had Christianity not taken over, somebody would eventually codify a school of though based on the 'old ways' of Rome - similarly, it doesn't seem unreasonable to me that this codification would become a religion/ideology/ethical system that gets adopted by the Roman Republic/Empire as a state doctrine.
So assuming this philosphical system comes into existence around the same time as Christianity, and as such competes with it, how do you think Rome would develop?
Alternatively, if you don't want to bother with Christianity you can have it come into existence earlier - maybe mid or late Republic era?