Christianity is not a Western religion, it is a Middle Eastern religion like Judaism, Islam, Baha'i, Zoroastrianism, etc. It spread from Palestine to Europe because Palestine was a Roman colony and its people were free to travel throughout the Roman Empire. As it grew stronger it sent monks to other countries to convert the local people.
The Buddhist monks traveled in the other direction. They converted the people of China, Japan, Korea, and southeastern Asia. They made a little westward progress but then Islam arose with its intolerance of diversity.
The fact that Buddhism thrives in the Orient while Christianity thrives in Europe and the Americas is largely an accident of history, the paths of the monks. However, there is also the matter of the natures of the two faiths. Christianity is an evangelical religion, like Islam, and instills in its people a sense of duty to go forth and convert the "heathens." Buddhism is not like that at all. Its spread throughout east Asia, for all of its success, was not belligerent. It does not threaten the unconverted with images of burning in hell and shun them to the margins of its society once it becomes dominant.
There is a sizeable segment of the American population that finds Buddhism to be quite sensible and attractive, but there are no Buddhist evangelists to act as its "salesmen." We are bombarded with Christian propaganda and accosted by Christian evangelists constantly. It's no wonder that people who just go with the flow and don't give a lot of thought to religion wind up as Christians rather than Buddhists.
More's the pity.
