Buddhism is not that appealing to those who don't come from a karmic-based culture. If you don't already accept the idea of reincarnation, then you don't worry about the threat of reincarnating as someone or something worse, or that the reincarnation cycle is itself something to be avoided. So why would you be interested in a solution to a problem you don't think exists? At best, it might appeal to a very narrow elite of over-educated philosophers.
When Buddhism did spread into non-karmic cultures, it did so in its variant of Mahayana Buddhism which did not yet exist when Asoka sent out his missionaries. Mahayana Buddhism introduced a lot of elements that made it very different from the proto-Theravada Buddhism of Asoka's time. Once you introduce an element that says, "There is this very great afterlife called a Pure Land, and if you are a good person, the Boddhisatva will take mercy on you and you will enjoy the afterlife", then you will get a very different response from people than, "After you die, you won't be reincarnated and can succumb to oblivion".
Why didn't Mahayana Buddhism spread to the West? By the time it was created, there was already a similar religion that promised people a good afterlife if they behaved well, it was called Christianity.