I found this interesting article/summary on the topic of Moses' Seat:
Eleazar L. Sukenik, in his important study published in 1934, Ancient Synagogues in Palestine and Greece, provided several examples of “Chairs of Moses” found by archaeologists. The one at Hammath-by-Tiberias is most interesting, because the back of the chair faces towards Jerusalem, picturing the law going forth from that direction as the synagogue audience is facing Zion. Stone seats positioned so that their occupant sat facing the congregation have been found in synagogues at Chorazin (cf. Matt. 11:21, Luke 10:13) in 1962, in En Gedi, and two Diaspora synagogues, (1) in Delos, the marble seat found in its ruins is probably the oldest example of a seat of Moses known (ca. 100 BC), and (2) Dura-Europos. Noel Rabbinowitz says this evidence taken together bolsters our conviction that the “Seat of Moses” was a physical seat upon which the Pharisees sat. He goes on to explain that most of the synagogue furniture was made of wood, which is why so few of these objects have survived.