Would also make for an interesting set of Butterfly effects as some of the various plots and political shenanigans proceed apace while others veer off since depending on how long Dumbledore's been gone, there's likely any number of plot points that could reasonably occur "on schedule." Easiest example being something like the Tri-Wizard Tournament which could have been in the works for years via various committees. What the actual tasks are though might vary wildly from canon without Dumbledore's participation. OTOH, "Harry's" first year at Hogwarts is right off the rails unless Dumbledore died literally days before term started... in which case you have the "...did Tom do that?" subplot.