I'm looking at clothing and dress from the period during the crusades and I suspect that buttons might have been something the crusaders took back with them from the Levant. They start popping up in the 12th and 13th century, yet I'm not sure because I cannot find any comprehensive book on buttons and their history (the Big Book of Buttons is only to be found in US libraries).
As you have noticed, buttons did not become popular as fasteners until around 1300. This is because before then clothing tended to consist of cloaks, robes, tunics and other loosely fitting garments that were easily secured with a pin (brooch or fibula). The Romans, Greeks and Levantines did wear buttons, but mostly as a sewn-on decoration, not as a fastener. They would sewn on gems, shells and many other types of button decorations. The exception was mostly for specialized equipment. For example, the Roman military boot, which was close fitting often was secured with a button.
The change that occurred around 1300 was that form-fitting tailoring developed. Shirts, trousers and coats were made to exactly fit the person. With this new type of clothing pins are impractical because you would need a lot of them. Therefore, buttons became popular and quickly spread along with the fashion for tailored clothing.