Why did most of the world adopt an alphabet-based writing system instead of a logographic writing system like Chinese characters?

The biggest advantage of alphabetic writing systems is that they are easy to learn. If you can learn just 26 characters, you can read and write any work in the English language, for example. Four year old children can and do learn all of the letters.

In a pictographic, logographic, or syllabaric writing system, there are always going to be many more characters that need to be learned. The more characters that exist, the more eduction is needed to learn them, and the more writing becomes the province of a special educated class.