does moses’ seat exist today?
The term “seat of Moses” or “chair of Moses” is not a reference to an actual chair. It is a reference to authority. It was a term used for the authority of the scribes and pharisees to interpret the Law and impose rules.
In Judaism a rabbi would preach sitting down with others gathered around him. Many times in the Gospels it recounts that Jesus sat down to teach and the people gathered around him. We see this symbolism still today when Bishops preach their homily seated in their presider’s chair and also the fact that in a cathedral only the Bishop uses the main presider’s chair, the *cathedra * (from which we get the word cathedral).
When the Pope issues a formal teaching infallibly it is said to come ex cathedra (from the chair). It is not meant to be a literal chair, but simply an ancient term that denotes authority.