We do see a unified church at the beginning though. We see the central authority at Jerusalem, with James, Peter, and John. Paul was accepted by this group, as both Acts and Paul tell us. Now, they do have disagreements, but they remain connected. The Jerusalem church remained behind Paul. At any time, they could have rejected him, but they didn't. So we see a united (or semi-united) front here.
Now, there were other groups, to a point. However, they were all under the same movement.
As for Paul's version winning out, not really. Yes, some of his ideas, actually many of them, stuck, but we see a variety of other ones were circumvented by later teachings, or even earlier teachings.
Now, Paul's teachings, in many instances, were in the same line of Jesus's. Paul expanded, and he directed his towards gentiles. But the basics were the same.
The reason why a gentile mission won out though was that it had to. A major reason why Christianity broke from Judaism was because of the destruction of the Temple, and the later redefining of Judaism. Judaism became centralized under Rabbinical Judaism (which was a logical move as centralizing gave them security in a time when their world was crashing down). With the centralization of Judaism, that meant that opposing views of Judaism were pushed away. Christianity was one of those opposing views that were pushed away. Being pushed away, it became natural for the movement to go to more of the gentile base.
And this movement that surfaced after the fall of the Temple, was Pauline in part, as he did work with the gentiles (as did other individuals in the movement). However, the Pauline ideas were influenced, and largely based off of the Jewish teachings of Jesus, as well as Judaism in general.
