I looked up that statuette from Sardinia. It was not found during a controlled archaeological excavation, so it’s not really possible to date it in definite terms. On stylistic grounds it is dated to between 900-300BC.
Just my idle speculation. I am inclined to think that what migt have been likely is this. There were the aulos-type instruments around the Mediterranean, them being local invention. When the Greeks arrived to the Northern shores of the Black Sea, they naturally took the instrument with them. The neighbouring peoples, Scythians and others, had a look and came up with a design that was more suited to their music, which would be the existing parallel double chanter with unequal fingering. At the same time around the South of the Mediterranean they could have independently arrived at a design that was suited to their style of music, which resulted in the parallel, equal-fingering chanter. Arab music has precious little use for polyphony even today, so that at least is plausible. On the oter hand the Cheremiss, Chuvash and other in the area are all polyphonic, and quite obviously home-grown, as there is no obvious influence from Russia, let alone further West in Europe. (Apart from a very small section that is Russian influenced, but it does form only a small part of the repertoire, and is quite separate from the rest.)