The way I've always seen it is that for a practical tailless arrangement that is stable, an airfoil with a positive pitching moment (1), or a wing with enough washout for a net positive pitching moment (2) or a swept wing arrangement with either a reflex airfoil (as in the first option above) or enough (less this time because of the moment arm from the wing sweep) washout to achieve a net positive pitching moment (3)......is a prerequisite, along with the proper relationship of center of gravity and neutral point (to balance the positive pitching moment) for a stable flying wing in pitch.The airfoil does not create stability in a flying wing or any other type of airplane. Stability comes from the relative placement of the neutral point (mostly due to planform) and the CG. The whole "stable airfoil" thing you see on all the R/C sites is a complete myth, reflecting a poor understanding of what's really going on. It's the difference between stability and trim.
Designing a flying wing is absolutely no different than designing any other kind of airplane. Just harder, because you have to get the wing to also do all the functions that would've been performed by the tail that's not there. Airplanes are a bunch of compromises flying in formation. Flying wings just require more - and harder - compromises.
As a prerequisite I see it as a rather important part of the total. Without it, the wing doesn't respond to pitch changes the way you need it to for pitch stability.
Calling it a "stable airfoil" is probably a misnomer when viewed literally. By itself, positive pitching moment is unstable too but it is a prerequisite for pitch stability in combination with the C.G./N.A. relationship.