In fact, I have seen this very debate play out in real life. We appointed a new President of the ASF during my term. That person had been a Director, but when they became President they stepped down from the Board.
For a non-profit charity that relies on volunteers, perfect separation is not always feasible, and especially in the early years when the ASF was smaller the President was often a Board member. But although I can't speak for anybody but myself, I believe that the ASF is likely to continue with an informal tradition of separating Board from Officers for the indefinite future.
A similar cultural change ought to be possible at Mozilla — and perhaps elsewhere.
The legal world has "managing partner" as the equivalent for LLPs.
It can vary, though; e.g. Kevin Sneader is the de facto CEO of McKinsey, but his official title is Global Managing Partner and his predecessor used Managing Director.
If they are not, the roles should be separate.
Their patron should rectify this, but it's possible they don't care.
The 'plan' may be to kill Firefox.
Is this really a problem? The CEO is often a director, and even as chairwoman I doubt anything in the bylaws prevents the board at large from removing her if they wanted to.
Instead, it seems like the board itself is the issue, since they allowed Mozilla to stray so far from its core mission.