My background in college was a double major of History and Religious Studies, but I've never found a satisfactory answer as to why Buddhism failed to develop in the West. It is known that Ashoka sent missionaries westward in the 3rd century B.C. This was a time before the other major proselytizing religions developed (Christianity, Islam), and Buddhism in OTL easily fit in with polytheistic traditions in Japan, China, and Southeast Asia thanks to its inclusive nature. The environment should have been perfect for it.
You would expect the western Greek philosophers to show some interest in Buddhism. It's not like Persian Zoroastrianism was immune to it either, since the 3rd century religion of Manichaeism was a fusion of Christianity, Buddhism, and Zoroastrianism. (I may do another topic on Manichaeism later. . .).
Why did Buddhism never take off in Europe and the Middle East?