In addition to Tamerlane, another example of successful government destruction of Christianity is Japan. In the late sixteenth century, Christianity was so successful there that people could plausibly anticipate that it would become a wholly Christian country within a couple of decades. By the early seventeenth century, Christians were being slaughtered so systematically that the religion was almost entirely wiped out. All that remained, when Japan re-opened to the west in the nineteenth century, were the Kakure Kirihorsehockyan, a small sect that had survived over the centuries but forgotten pretty much everything about their supposedly Catholic religion. They had no doctrine of the Trinity, rejected the authority of Rome, and all their images of Jesus looked like the Buddha.
On Constantine, Lactantius gives a different account of his conversion from Eusebius' more well-known one. Either way, both were surely telescoping a longer process into a simple story. However, I don't think the details of how Constantine converted are really important; what matters is what he did. (Which, in the first instance, wasn't all that much, as Christianity was already de facto legal in the western empire at that time, thanks partly to Constantine's father.)