WTP
If the father is a contract employee of the synagogue there is nothing to inherent. Personal services contracts dies with the person. In fact it is quite common for the rabbi’s employment contract to have a set term. The synagogue is not forced to renew the contract unless life tenure has been granted.
This is far different than shtieblach that may operate in the home of the rabbi. There the son may well take over, but congregants are free to leave and not support the son.