It makes no sense to have an American “embassy” to the Roman Catholic Church, unless the U.S. is going to have an “embassy” to whatever HQ there is to any other major religious denomination.
The embassy isn’t to the Church per se but to the Vatican City State, which is all that is left of the Papal States—109 acres that are a sovereign nation, and not part of Italy. It is much diminished, but the nation has existed in one way or another for 1257 years (see my post 35 above). If the head of another denomination also happened to be head of a state, it would make sense to have an ambassador to that state.
The Pope does pack some additional punch as well, usually being well-respected, the spiritual leader of more than a billion, and alleged to have good connections on high, all of which are reasons to give more weight to the position than to ambassadorships to other very small states.