I think you have an accurate picture there. But do you really think any of it is due to "good fortune?"
I think that most of it was due to good fortune. I don't think that Christianity had any direct connection to the voyages of discovery and with the scientific and industrial revolutions, for example, but it certainly benefited from effectively being a free-rider on growing European global ascendency. That's what spread Christianity around the world and why it's all over North and South America and in places like the Philippines today.
Imagine what would have happened if it was China that had sent its ships all around the world instead. They almost did, organizing a big exploratory fleet in the Indian ocean in the early Ming period (early 1400s CE) that visited the Persian gulf and sailed as far as Africa. But a later Ming emperor decided that the whole thing was a dangerous waste of money and outlawed any Chinese use of oceangoing ships without a hard-to-get government approval. ... Then a few decades later, the Europeans started sailing into their ports and we all know how that turned out.
If it had been the Chinese that sailed into European ports, if it had been China that conquered the Americas and Australia and South Africa and every small island in every ocean... We'd probably all be Confucians, Daoists and Buddhists today, writing internet posts about how those Asian religions are innately superior to the crude Western superstitions.
It seems to me that Constantine accepted it because it was successful and he needed it to weld power.
There's a whole historical literature about what motivated Constantine. Part of our problem with him is that we don't know a lot about the details of his thinking, apart from Christian writers like Eusebius who wanted to spin the account of Constantine's motivations in ways that would subsequently benefit the church.
There's one theory that he was basically a secular individual who saw that Diocletian's attempts to stamp out Christianity weren't working, and tried to coopt and use it instead. Another theory argues that Constantine was a committed Christian himself and favoring the church reflected his own belief.
I'm inclined to kind of straddle that difference. We know that Constantine was raised a pagan, but a rather unconventional pagan in a very eclectic, inclusive and syncretistic polytheistic atmosphere. We know that there were Christians among those who taught him as a youth, so he'd definitely been exposed to the religion.
My tendency is to read Constantine as a deep-down polytheist, somebody who kind of implicitly assumed that there are lots of gods (or at least lots of personas of one totally transcendent godhead, a quasi-monotheistic idea that was already very common anong Greco-Roman pagans) and that one offered cult, sacrificing to and offering prayers to and patronizing, whatever god was most auspicious to that individual in that circumstance. (Polythiests rarely actively worship all gods, they choose particular gods as theirs without denying the others.)
Constantine seems to have favored a Mithraic-style sun god in his younger years but seems to have started honoring the Christian god around 313 CE or so. There's a later story about how the Christian god led him to victory in battle at the Milvian bridge amidst glorious miraculous appearances in the sky. I don't give the miracle tales any credence, but it's certainly possible that an anxious general Constantine was offering prayers to different gods and happened to win his victory after praying to the Christian one. So he might well have decided, pagan style, that the Christian god was indeed the auspicious god for him to continue favoring.
For some years after, Constantine still seems to have recognized additional gods and still issued coins with his likeness and non-Christian inscriptions. But he gradually seems to have become more of a conventionally devout Christian believer as time passed and he found himself more and more closely linked to the Christian ecclesiastical hierarchy. That relationship quickly became symbiotic since he saw them as the nucleus of a unifying state-church that the emperor would naturally dominate as a caesaro-papistic divinely-chosen monarch, while the bishops saw imperial favor as their own pathway to legal recognition, earthly power and lavish funding.
It's kind of amazing to me how Constantine presided over the Council of Nicea with great power and pomp, pushing the church to adopt the orthodox doctrine of the trinity, despite the fact that he hadn't even been baptized a Christian at that point. No doubt some of the church's miracle stories about Constantine's earlier heavenly vision were intended to paper over that problem, showing that God had indeed chosen and elected him despite his not being a conventional baptised Christian. And Constantine doubtless had no objection to hagiography that turned him into a demigod, something that Roman emperors had long wanted to be.
Also, that it arose and survived because it was superior to the old polytheisms that it replaced.
I don't think that's true. In some ways, yes, but in other ways no. The old paganism might easily have survived had things gone differently. We see examples of what might have resulted today in Indian and Chinese religion.
Hinduism in India isn't unlike the old Greco-Roman syncretistic religion, with its superficial polytheism and its deeper and more monistic philosophical supports. Hinduism is more psychologized than Greek and Roman religion because of the influence of its yogic meditation traditions and it's dramatically different in terms of the caste system.
Traditional Chinese folk religion with its many local gods and its Daoist alchemy and magic isn't dissimilar from the more street-level aspects of the old Greco-Roman paganism either, with Confucianism kind of riding on top among the educated intellectuals like Greek philosophy did further West.
