Speaking from my experience as player (or mod) in boards where the owner and the admins weren't around, IMHO, they aren't annoying as long as they aren't the OOC type and more especially the God type.
Problems come when difficult players drop around the board (the troll types, or to hack the site, etc), since someone should have granted admin powers. But if this doesn't happen, well, people can play (almost) the same, having the mods "temporary" acting as admins. But I agree, it is annoying since you can't do a lot of things.
But in the end, if the problems persist for months, the best thing to do is to create another board and to move there, yep.
Nolen.
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make sure they are trustworthy. Lots of complaints in help forums are also about moderators gone bad who deleted posts and such
Oh, yes, definitely yes. And to delete the whole layout and settings and all.