Problem is, while that might indeed kill Christianity, it would probably just mean that Constantine would slot Mithraism into his Imperial domination schemes instead.
In the late 200s AD there were two mostly-underground monotheistic mystery cults rapidly gaining adherents in the Roman Empire. There were a lot of similarities between the two, at least superficially. For example, there was a lot of emphasis on communal ritual meals. One was Christianity. The other was Mithraism. Constantine was intrigued by both. We know he was involved in Mithraism in his youth.
But what Constantine really liked the idea of using religion to unify the Roman Empire. By the 300s, the Roman Empire was beginning to fragment, with regular civil wars. Constantine came to power in one of those civil wars. He thought that if everyone worshiped the same god (instead of different gods worshiped in different places, with the Roman pantheon and emperors as a thin veneer of unity), it would help keep the whole ramshackle edifice together. (Spoiler alert: it did not.) So he picked one of the two monotheistic religions that was rapidly gaining in popularity, and encouraged people to convert to it, heaping power and wealth on (some of) them. And that's how Christianity became an imperial religion.
Christianity changed rapidly in response to that. Major parts of the religion were changed or dropped entirely. For example, until Constantine, the vast majority of Christians were strict pacifists. In most communities, soldiers were required to leave the army and find a new trade before they could be baptized. Obviously, this was unacceptable if Christianity was going to become the religion of the Roman Empire. In a straight-up choice between pacifism and Imperial power, the Christian church as a whole dropped the pacifism like a hot potato. 100 years after Constantine you have St. Augustine laying out the "Just War" theory where war is fine as long as you have a good reason for it. That's a complete 180 from everything the early Christians believed. There are many other examples of things that got dropped or changed in Christianity to make it more palatable to Imperial might.
There are a lot of toxic things in Christianity as we know it. But the thing is ... many of them come from this process of adapting their beliefs and practices to fit what Constantine (and later Emperors, and the entire power structure of the Empire) wanted Christianity to be. Namely, something tame that affirmed and enforced the existing Imperial power structure. And Christianity has been a partner and tool of the power structures of the dominant culture ever since. This is one of the reasons there's so much difference between Jesus' teachings and Christian teachings, in so many cases. In a straight-up choice between faithfulness and power ... a majority of Christians in the last two thousand years have most often chosen power.
But here's the thing. If Christianity didn't exist, that doesn't mean none of this would have happened. It just means that Constantine would probably have chosen Mithraism instead. Do you think the Mithraists would have been any less willing to take the power and wealth on offer to them, in exchange for becoming a lackey of empire? Do you think Christianity was uniquely corruptible? I don't.