It's not that the Speaker is not a Member, it's that the order of succession lists the Speaker of the House in the line of succession. In the case of the inability of the President and Vice President to continue to serve, that would mean an unelected person (except by the House Membership) would become President, a prospect that a number of people object to. In all other cases, the person has been elected to office, or appointed and confirmed by a majority of the Senate - thus vetted publicly. I was merely pointing out that there are expectations of the Speakership that are not, strictly speaking, in the Constitution.
There are practical reasons, I think, for both selection of the Speaker from the membership (the norm), and for picking an outside person. Under the present situation, the Speaker acts not only as the leader of the House itself, but of their party. I consider this, often, to be a conflict of interest, and many Speakers have taken that to extremes (Newt Gingrich and Denny Hastert come to mind). The House is supposed to represent all of the people, not just their party, and the Speaker should be someone that takes that to heart. In that regard, selection of someone from outside of the Members could further that purpose. On the other hand, I think it would be difficult, as a practical matter, for the Speaker to influence Members if they, themselves, didn't have to run for office or have a vote in Congress.