Oh here we go. I've been a member up to a second in command left in charge because the owner just dropped off. Maybe they return, maybe they never will.
Why would I stay? Generally because there's enough of a community and/or content and a buffered enough staff group which can still function that you don't really notice. A community has taken off when it can survive absentee ownership, even though that leads to numerous deep running problems. If lucky it has people with technical access who can prod it along, a lease on life especially if stuff like the domain does keep getting renewed and someone kicks the box when it coughs. However, and I suspect this applies to a majority of boards that pass through a place like this, the absence of the owner probably means death either sudden or gradual and at that point it's just time to move on. If there's any future depends if the owner has made the community solvent enough to persist beyond their absence.
A best case scenario is that they let go, and pass the torch to a willing successor.
Now here is a question... would you rather a community in limbo have a final answer (closure, in both senses) because the owner is gone, or would you rather it stay online and exist for whatever value its content or surviving community can come up with? This assumes the owner will not sell or transfer, but is hanging on - perhaps appearing very rarely, perhaps unable to let go despite it not necessarily being profitable and slowly, gradually, slipping away into obscurity in its niche. Add a touch of 'open a few decades' and having once been a leader, even the leader in its market.