>you have to be House Rep to be nominated as Speaker
Regardless, the Speaker—who has always been (but is not required to be) a House Member with the same obligations to his or her constituents like the other 434 Members—is at the levers of power. The Speaker is simultaneously the House’s presiding officer, party leader, and the institution’s administrative head, among other duties.
The Speaker is elected at the beginning of a new Congress by a majority of the Representatives-elect from candidates separately chosen by the majority- and minority-party caucuses. These candidates are elected by their party members at the organizing caucuses held soon after the new Congress is elected. In cases of an unexpected vacancy during a Congress a new Speaker is elected by a majority of the House from candidates previously chosen by the two parties.
https://history.house.gov/People/Office/Speakers-Intro/