rav yosef chaim zonenfeld ztz”l founded the Eida Charedis, who was the first Gaved. After his passing, the Eida begged the Satmar Rebbe ztz”l to become the second Gaved, but he declined (He ultimately conceded after the war and lead the Jerusalem community until his passing in the 70s).
The eida was and continues to be very much a unified beis din for the ashkenazy kehila, both litvish chasidish and otherwise, but its roots are (also) very Hungarian. So, I’m not sure where you got your history from.