Even in polytheist Rome, once Lepidus bit the dust, the Emperor was always also the high priest.
I, myself, finds to rather agree with Paul Veynes when he argues that Christianity had a genuine spiritual attraction compared to Roman-Hellenistic beliefs. It's why I want to stress that the reasons Christianity did blossomed in medieval Europe aren't systematically structural, in the sense of blunt and concious self-interest : ideologies, beliefs, doesn't spread this way, not only. It's why imperial cults as Sol Invictus didn't really worked out IMO, and why Christianity (which was an imperial cult, eventually, rather than a popular religion adopted by the emperor) did.My question is a bit off-topic (since your post focussed on the time after 500 CE): if I understand your statement, Christianity had no advantage over Hellenistic paganism.
If you allow me to quote.
More than evangelic spirit, that spiritual culture or exalthation yet to come of a suffering Christ and Holy Mother (Byzantium and St. Bernard are still to happen), others attracts, unknown from traditional paganism, were enough to provoke most of conversions : loving piety of a religion of love, collective fervours during long weekly synaxes of a communautary cult, hope and joy of a supernatural destination, peace of soul different from stoician atraxis, and before anything the "bourgeois" moralism that german historians underline; there was, or as we can believe, a certain moralism among respectable plebs media. All of this could only comfort representants of public authority, if they deemed worth to inform themselves.
He mades at some point an interesting comparison with socialism, in the sense that an universalist belief first propaged by intellectual circles and semi-elites, eventually managed to get down in numbers to the lower classes, not because these lites were self-interested (for political control, machiavellian devices or muhahahahaha evilness), but because it appealed to them in the general cultural and ideological crisis, resonating with their own zeitgast, both practically and culturally.
Paul Veynes sees as well a crisis of paganism, among cultivated and scholarly elites, caused by the lack of practical and spiritual cohesion between philosophic and civic religion.
He does stress that among the general population, paganism as a custom was entranched and could have continued, but not so much among scholars, philosophers and general elites.
Now, he certainly doesn't say that the Late Roman empire was pregnant with Christianity and, fairly IMO, points that it was both the interest and probably more the genuine faith of Constantin that propelled a fairly (if significant) minor religion to a main one. But, structually, it was obviously a fitting choice, more than the "natural" evolution of traditional paganism so far, so to speak.
Which is kinda the point, altough @clem attlee would be far more able than I am to dissert on the imperial religious structures and dynamics, and he seems to point that late Roman religion had institutional autonomy and unity of its own. The strength of Christianity comes, at least when it comes early medieval era, that it was partially separated from politics (up to a point of course, the "natural" ruler of the churches remain the king, most of the time), and that a king adopting Christianity didn't have to bend to the king of Franks or Rome if he could avoid it, and at the contrary could structurate its own clergy in time (it could take more or less of this, arguably, but the HRE never really obtained a protectorate trough religion on Scandinavia, for exemple)Hellenism too was a closed system which peneprated civic and religious life. In Greek (and, for that matter, Roman) cities, there was no separation of politics and religion. The Gods and their cult were omnipresent.